Properties

Label 40-65e20-1.1-c1e20-0-0
Degree $40$
Conductor $1.812\times 10^{36}$
Sign $1$
Analytic cond. $2.01284\times 10^{-6}$
Root an. cond. $0.720435$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 2·3-s + 11·4-s + 12·6-s − 2·7-s + 6·8-s + 8·9-s − 16·11-s − 22·12-s − 4·13-s + 12·14-s − 45·16-s + 4·17-s − 48·18-s − 20·19-s + 4·21-s + 96·22-s − 10·23-s − 12·24-s + 9·25-s + 24·26-s − 8·27-s − 22·28-s + 30·32-s + 32·33-s − 24·34-s + 88·36-s + ⋯
L(s)  = 1  − 4.24·2-s − 1.15·3-s + 11/2·4-s + 4.89·6-s − 0.755·7-s + 2.12·8-s + 8/3·9-s − 4.82·11-s − 6.35·12-s − 1.10·13-s + 3.20·14-s − 11.2·16-s + 0.970·17-s − 11.3·18-s − 4.58·19-s + 0.872·21-s + 20.4·22-s − 2.08·23-s − 2.44·24-s + 9/5·25-s + 4.70·26-s − 1.53·27-s − 4.15·28-s + 5.30·32-s + 5.57·33-s − 4.11·34-s + 44/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(5^{20} \cdot 13^{20}\)
Sign: $1$
Analytic conductor: \(2.01284\times 10^{-6}\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{65} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 5^{20} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.002070078433\)
\(L(\frac12)\) \(\approx\) \(0.002070078433\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 9 T^{2} + 33 T^{4} + 16 T^{5} - 24 T^{6} - 48 T^{7} - 906 T^{8} - 96 T^{9} + 7346 T^{10} - 96 p T^{11} - 906 p^{2} T^{12} - 48 p^{3} T^{13} - 24 p^{4} T^{14} + 16 p^{5} T^{15} + 33 p^{6} T^{16} - 9 p^{8} T^{18} + p^{10} T^{20} \)
13 \( 1 + 4 T - 5 T^{2} - 64 T^{3} - 63 T^{4} + 116 T^{5} + 1124 T^{6} + 908 p T^{7} + 37062 T^{8} - 102788 T^{9} - 974094 T^{10} - 102788 p T^{11} + 37062 p^{2} T^{12} + 908 p^{4} T^{13} + 1124 p^{4} T^{14} + 116 p^{5} T^{15} - 63 p^{6} T^{16} - 64 p^{7} T^{17} - 5 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
good2 \( 1 + 3 p T + 25 T^{2} + 39 p T^{3} + 101 p T^{4} + 111 p^{2} T^{5} + 845 T^{6} + 177 p^{3} T^{7} + 2105 T^{8} + 357 p^{3} T^{9} + 29 p^{7} T^{10} + 159 p^{5} T^{11} + 7953 T^{12} + 6861 p T^{13} + 23753 T^{14} + 18951 p T^{15} + 26915 p T^{16} + 261 p^{8} T^{17} + 73281 T^{18} + 9567 p^{3} T^{19} + 92661 T^{20} + 9567 p^{4} T^{21} + 73281 p^{2} T^{22} + 261 p^{11} T^{23} + 26915 p^{5} T^{24} + 18951 p^{6} T^{25} + 23753 p^{6} T^{26} + 6861 p^{8} T^{27} + 7953 p^{8} T^{28} + 159 p^{14} T^{29} + 29 p^{17} T^{30} + 357 p^{14} T^{31} + 2105 p^{12} T^{32} + 177 p^{16} T^{33} + 845 p^{14} T^{34} + 111 p^{17} T^{35} + 101 p^{17} T^{36} + 39 p^{18} T^{37} + 25 p^{18} T^{38} + 3 p^{20} T^{39} + p^{20} T^{40} \)
3 \( 1 + 2 T - 4 T^{2} - 16 T^{3} + T^{4} + 56 T^{5} + 40 T^{6} - 50 p T^{7} - 176 T^{8} + 406 T^{9} + 640 T^{10} - 1360 T^{11} - 2741 T^{12} + 1196 p T^{13} + 9092 T^{14} - 11098 T^{15} - 32293 T^{16} + 42280 T^{17} + 47704 p T^{18} - 19340 p T^{19} - 519968 T^{20} - 19340 p^{2} T^{21} + 47704 p^{3} T^{22} + 42280 p^{3} T^{23} - 32293 p^{4} T^{24} - 11098 p^{5} T^{25} + 9092 p^{6} T^{26} + 1196 p^{8} T^{27} - 2741 p^{8} T^{28} - 1360 p^{9} T^{29} + 640 p^{10} T^{30} + 406 p^{11} T^{31} - 176 p^{12} T^{32} - 50 p^{14} T^{33} + 40 p^{14} T^{34} + 56 p^{15} T^{35} + p^{16} T^{36} - 16 p^{17} T^{37} - 4 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
7 \( 1 + 2 T - 6 p T^{2} - 68 T^{3} + 929 T^{4} + 1124 T^{5} - 14194 T^{6} - 1598 p T^{7} + 168832 T^{8} + 9642 p T^{9} - 1684962 T^{10} - 185940 T^{11} + 2130957 p T^{12} - 472932 T^{13} - 121949582 T^{14} + 5345154 T^{15} + 943522539 T^{16} - 7858360 T^{17} - 6992208836 T^{18} - 30581128 T^{19} + 49860115936 T^{20} - 30581128 p T^{21} - 6992208836 p^{2} T^{22} - 7858360 p^{3} T^{23} + 943522539 p^{4} T^{24} + 5345154 p^{5} T^{25} - 121949582 p^{6} T^{26} - 472932 p^{7} T^{27} + 2130957 p^{9} T^{28} - 185940 p^{9} T^{29} - 1684962 p^{10} T^{30} + 9642 p^{12} T^{31} + 168832 p^{12} T^{32} - 1598 p^{14} T^{33} - 14194 p^{14} T^{34} + 1124 p^{15} T^{35} + 929 p^{16} T^{36} - 68 p^{17} T^{37} - 6 p^{19} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
11 \( 1 + 16 T + 140 T^{2} + 888 T^{3} + 4581 T^{4} + 20232 T^{5} + 78492 T^{6} + 266032 T^{7} + 70212 p T^{8} + 1836640 T^{9} + 3047260 T^{10} - 144872 T^{11} - 28959109 T^{12} - 162089992 T^{13} - 665171868 T^{14} - 2473230160 T^{15} - 9110751105 T^{16} - 33941309200 T^{17} - 124740924712 T^{18} - 443369710480 T^{19} - 1505615976312 T^{20} - 443369710480 p T^{21} - 124740924712 p^{2} T^{22} - 33941309200 p^{3} T^{23} - 9110751105 p^{4} T^{24} - 2473230160 p^{5} T^{25} - 665171868 p^{6} T^{26} - 162089992 p^{7} T^{27} - 28959109 p^{8} T^{28} - 144872 p^{9} T^{29} + 3047260 p^{10} T^{30} + 1836640 p^{11} T^{31} + 70212 p^{13} T^{32} + 266032 p^{13} T^{33} + 78492 p^{14} T^{34} + 20232 p^{15} T^{35} + 4581 p^{16} T^{36} + 888 p^{17} T^{37} + 140 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \)
17 \( 1 - 4 T + 65 T^{2} - 172 T^{3} + 1418 T^{4} - 2228 T^{5} + 8655 T^{6} + 1808 T^{7} - 268746 T^{8} + 1272808 T^{9} - 13137085 T^{10} + 51346880 T^{11} - 296572088 T^{12} + 838147728 T^{13} - 3153368693 T^{14} + 3609124160 T^{15} + 11120075953 T^{16} - 145288002868 T^{17} + 1472996590450 T^{18} - 5409253581872 T^{19} + 36582230959020 T^{20} - 5409253581872 p T^{21} + 1472996590450 p^{2} T^{22} - 145288002868 p^{3} T^{23} + 11120075953 p^{4} T^{24} + 3609124160 p^{5} T^{25} - 3153368693 p^{6} T^{26} + 838147728 p^{7} T^{27} - 296572088 p^{8} T^{28} + 51346880 p^{9} T^{29} - 13137085 p^{10} T^{30} + 1272808 p^{11} T^{31} - 268746 p^{12} T^{32} + 1808 p^{13} T^{33} + 8655 p^{14} T^{34} - 2228 p^{15} T^{35} + 1418 p^{16} T^{36} - 172 p^{17} T^{37} + 65 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \)
19 \( 1 + 20 T + 8 p T^{2} + 364 T^{3} - 1823 T^{4} - 13572 T^{5} - 15744 T^{6} + 58980 T^{7} - 559020 T^{8} - 6484036 T^{9} - 18760472 T^{10} + 16033220 T^{11} + 127241991 T^{12} - 326018372 T^{13} + 1166119272 T^{14} + 35902407492 T^{15} + 114702291975 T^{16} - 403614272520 T^{17} - 2514081037864 T^{18} + 7754137461800 T^{19} + 90038724597720 T^{20} + 7754137461800 p T^{21} - 2514081037864 p^{2} T^{22} - 403614272520 p^{3} T^{23} + 114702291975 p^{4} T^{24} + 35902407492 p^{5} T^{25} + 1166119272 p^{6} T^{26} - 326018372 p^{7} T^{27} + 127241991 p^{8} T^{28} + 16033220 p^{9} T^{29} - 18760472 p^{10} T^{30} - 6484036 p^{11} T^{31} - 559020 p^{12} T^{32} + 58980 p^{13} T^{33} - 15744 p^{14} T^{34} - 13572 p^{15} T^{35} - 1823 p^{16} T^{36} + 364 p^{17} T^{37} + 8 p^{19} T^{38} + 20 p^{19} T^{39} + p^{20} T^{40} \)
23 \( 1 + 10 T + 104 T^{2} + 760 T^{3} + 4021 T^{4} + 25240 T^{5} + 110596 T^{6} + 629514 T^{7} + 3599888 T^{8} + 16216374 T^{9} + 101637156 T^{10} + 372526544 T^{11} + 1815379751 T^{12} + 8866821764 T^{13} + 37211800024 T^{14} + 311193036366 T^{15} + 1423456049115 T^{16} + 8723716465064 T^{17} + 42970194226792 T^{18} + 170399078825124 T^{19} + 1023524571922080 T^{20} + 170399078825124 p T^{21} + 42970194226792 p^{2} T^{22} + 8723716465064 p^{3} T^{23} + 1423456049115 p^{4} T^{24} + 311193036366 p^{5} T^{25} + 37211800024 p^{6} T^{26} + 8866821764 p^{7} T^{27} + 1815379751 p^{8} T^{28} + 372526544 p^{9} T^{29} + 101637156 p^{10} T^{30} + 16216374 p^{11} T^{31} + 3599888 p^{12} T^{32} + 629514 p^{13} T^{33} + 110596 p^{14} T^{34} + 25240 p^{15} T^{35} + 4021 p^{16} T^{36} + 760 p^{17} T^{37} + 104 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \)
29 \( 1 + 117 T^{2} + 6870 T^{4} - 8328 T^{5} + 9015 p T^{6} - 908016 T^{7} + 6368106 T^{8} - 49449240 T^{9} + 86554363 T^{10} - 1654451928 T^{11} - 517414740 T^{12} - 27891514632 T^{13} - 83953593117 T^{14} + 350011028904 T^{15} - 3978002845539 T^{16} + 1543536257664 p T^{17} - 154268740846974 T^{18} + 1909124569268712 T^{19} - 4952841504137748 T^{20} + 1909124569268712 p T^{21} - 154268740846974 p^{2} T^{22} + 1543536257664 p^{4} T^{23} - 3978002845539 p^{4} T^{24} + 350011028904 p^{5} T^{25} - 83953593117 p^{6} T^{26} - 27891514632 p^{7} T^{27} - 517414740 p^{8} T^{28} - 1654451928 p^{9} T^{29} + 86554363 p^{10} T^{30} - 49449240 p^{11} T^{31} + 6368106 p^{12} T^{32} - 908016 p^{13} T^{33} + 9015 p^{15} T^{34} - 8328 p^{15} T^{35} + 6870 p^{16} T^{36} + 117 p^{18} T^{38} + p^{20} T^{40} \)
31 \( 1 - 104 T^{3} + 1794 T^{4} - 3320 T^{5} + 5408 T^{6} - 552640 T^{7} + 1315037 T^{8} + 1490240 T^{9} + 53283808 T^{10} - 779482432 T^{11} + 2015402168 T^{12} + 9179164672 T^{13} + 126121189280 T^{14} - 557942017984 T^{15} - 69210310750 T^{16} - 4465408437184 T^{17} + 158380681056 p^{2} T^{18} - 15988767100848 p T^{19} - 2929717194662260 T^{20} - 15988767100848 p^{2} T^{21} + 158380681056 p^{4} T^{22} - 4465408437184 p^{3} T^{23} - 69210310750 p^{4} T^{24} - 557942017984 p^{5} T^{25} + 126121189280 p^{6} T^{26} + 9179164672 p^{7} T^{27} + 2015402168 p^{8} T^{28} - 779482432 p^{9} T^{29} + 53283808 p^{10} T^{30} + 1490240 p^{11} T^{31} + 1315037 p^{12} T^{32} - 552640 p^{13} T^{33} + 5408 p^{14} T^{34} - 3320 p^{15} T^{35} + 1794 p^{16} T^{36} - 104 p^{17} T^{37} + p^{20} T^{40} \)
37 \( 1 + 4 T - 255 T^{2} - 628 T^{3} + 35070 T^{4} + 42996 T^{5} - 3363009 T^{6} - 1430592 T^{7} + 250169718 T^{8} - 2742320 T^{9} - 15589359357 T^{10} + 2630094216 T^{11} + 857793078820 T^{12} - 140911146544 T^{13} - 42819935288493 T^{14} + 4890972627376 T^{15} + 1957545276095489 T^{16} - 128213364289956 T^{17} - 82094063860519094 T^{18} + 1723066089909848 T^{19} + 3164986030262490540 T^{20} + 1723066089909848 p T^{21} - 82094063860519094 p^{2} T^{22} - 128213364289956 p^{3} T^{23} + 1957545276095489 p^{4} T^{24} + 4890972627376 p^{5} T^{25} - 42819935288493 p^{6} T^{26} - 140911146544 p^{7} T^{27} + 857793078820 p^{8} T^{28} + 2630094216 p^{9} T^{29} - 15589359357 p^{10} T^{30} - 2742320 p^{11} T^{31} + 250169718 p^{12} T^{32} - 1430592 p^{13} T^{33} - 3363009 p^{14} T^{34} + 42996 p^{15} T^{35} + 35070 p^{16} T^{36} - 628 p^{17} T^{37} - 255 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
41 \( 1 - 10 T + 53 T^{2} + 370 T^{3} - 5474 T^{4} + 29582 T^{5} + 60499 T^{6} - 2473998 T^{7} + 17915882 T^{8} - 56078982 T^{9} - 425519157 T^{10} + 5305982318 T^{11} - 21403552108 T^{12} - 67497771986 T^{13} + 1513554719563 T^{14} - 7387733690454 T^{15} + 7177932842429 T^{16} + 156654968651636 T^{17} - 19723383509582 p T^{18} - 3491954639048348 T^{19} + 39411740829335628 T^{20} - 3491954639048348 p T^{21} - 19723383509582 p^{3} T^{22} + 156654968651636 p^{3} T^{23} + 7177932842429 p^{4} T^{24} - 7387733690454 p^{5} T^{25} + 1513554719563 p^{6} T^{26} - 67497771986 p^{7} T^{27} - 21403552108 p^{8} T^{28} + 5305982318 p^{9} T^{29} - 425519157 p^{10} T^{30} - 56078982 p^{11} T^{31} + 17915882 p^{12} T^{32} - 2473998 p^{13} T^{33} + 60499 p^{14} T^{34} + 29582 p^{15} T^{35} - 5474 p^{16} T^{36} + 370 p^{17} T^{37} + 53 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \)
43 \( 1 - 10 T - 40 T^{2} + 1364 T^{3} - 3983 T^{4} - 77956 T^{5} + 640228 T^{6} + 1596326 T^{7} - 38848928 T^{8} + 76430850 T^{9} + 1212163500 T^{10} - 6607056052 T^{11} - 135285871 p T^{12} + 76098476592 T^{13} - 657626537288 T^{14} + 14318581328546 T^{15} - 47723084208453 T^{16} - 1005173616928256 T^{17} + 8200326933037552 T^{18} + 21221387803082188 T^{19} - 490364809303665792 T^{20} + 21221387803082188 p T^{21} + 8200326933037552 p^{2} T^{22} - 1005173616928256 p^{3} T^{23} - 47723084208453 p^{4} T^{24} + 14318581328546 p^{5} T^{25} - 657626537288 p^{6} T^{26} + 76098476592 p^{7} T^{27} - 135285871 p^{9} T^{28} - 6607056052 p^{9} T^{29} + 1212163500 p^{10} T^{30} + 76430850 p^{11} T^{31} - 38848928 p^{12} T^{32} + 1596326 p^{13} T^{33} + 640228 p^{14} T^{34} - 77956 p^{15} T^{35} - 3983 p^{16} T^{36} + 1364 p^{17} T^{37} - 40 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \)
47 \( ( 1 + 20 T + 486 T^{2} + 6548 T^{3} + 93533 T^{4} + 960312 T^{5} + 10222280 T^{6} + 85662136 T^{7} + 747535234 T^{8} + 5352082240 T^{9} + 40243643940 T^{10} + 5352082240 p T^{11} + 747535234 p^{2} T^{12} + 85662136 p^{3} T^{13} + 10222280 p^{4} T^{14} + 960312 p^{5} T^{15} + 93533 p^{6} T^{16} + 6548 p^{7} T^{17} + 486 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
53 \( 1 + 10 T + 50 T^{2} + 608 T^{3} + 8911 T^{4} + 42256 T^{5} + 161842 T^{6} + 1379418 T^{7} + 11864269 T^{8} + 42078400 T^{9} + 116867400 T^{10} - 342809968 T^{11} + 2993373476 T^{12} + 82242842224 T^{13} + 232250742760 T^{14} - 4408284842096 T^{15} - 104445179440542 T^{16} - 7400825808492 p T^{17} - 1433330906377060 T^{18} - 34029241052501104 T^{19} - 545288959843663782 T^{20} - 34029241052501104 p T^{21} - 1433330906377060 p^{2} T^{22} - 7400825808492 p^{4} T^{23} - 104445179440542 p^{4} T^{24} - 4408284842096 p^{5} T^{25} + 232250742760 p^{6} T^{26} + 82242842224 p^{7} T^{27} + 2993373476 p^{8} T^{28} - 342809968 p^{9} T^{29} + 116867400 p^{10} T^{30} + 42078400 p^{11} T^{31} + 11864269 p^{12} T^{32} + 1379418 p^{13} T^{33} + 161842 p^{14} T^{34} + 42256 p^{15} T^{35} + 8911 p^{16} T^{36} + 608 p^{17} T^{37} + 50 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 + 16 T + 320 T^{2} + 3664 T^{3} + 37117 T^{4} + 331488 T^{5} + 2103744 T^{6} + 16380920 T^{7} + 100348484 T^{8} + 883337776 T^{9} + 9675056288 T^{10} + 81687354080 T^{11} + 809702972683 T^{12} + 5715721431720 T^{13} + 35547465831840 T^{14} + 241780218876136 T^{15} + 1057616949186407 T^{16} + 10092819361240512 T^{17} + 89215361760986816 T^{18} + 721667655155172056 T^{19} + 7602396017918841272 T^{20} + 721667655155172056 p T^{21} + 89215361760986816 p^{2} T^{22} + 10092819361240512 p^{3} T^{23} + 1057616949186407 p^{4} T^{24} + 241780218876136 p^{5} T^{25} + 35547465831840 p^{6} T^{26} + 5715721431720 p^{7} T^{27} + 809702972683 p^{8} T^{28} + 81687354080 p^{9} T^{29} + 9675056288 p^{10} T^{30} + 883337776 p^{11} T^{31} + 100348484 p^{12} T^{32} + 16380920 p^{13} T^{33} + 2103744 p^{14} T^{34} + 331488 p^{15} T^{35} + 37117 p^{16} T^{36} + 3664 p^{17} T^{37} + 320 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 + 16 T - 291 T^{2} - 4624 T^{3} + 62614 T^{4} + 791392 T^{5} - 10200029 T^{6} - 91274480 T^{7} + 1347937546 T^{8} + 7746824480 T^{9} - 143223609453 T^{10} - 467039401632 T^{11} + 12554800039532 T^{12} + 18247567668448 T^{13} - 918360371027813 T^{14} - 202692877277696 T^{15} + 58440043341660413 T^{16} - 20244855642251520 T^{17} - 3426930072683677342 T^{18} + 853407459595356016 T^{19} + \)\(20\!\cdots\!68\)\( T^{20} + 853407459595356016 p T^{21} - 3426930072683677342 p^{2} T^{22} - 20244855642251520 p^{3} T^{23} + 58440043341660413 p^{4} T^{24} - 202692877277696 p^{5} T^{25} - 918360371027813 p^{6} T^{26} + 18247567668448 p^{7} T^{27} + 12554800039532 p^{8} T^{28} - 467039401632 p^{9} T^{29} - 143223609453 p^{10} T^{30} + 7746824480 p^{11} T^{31} + 1347937546 p^{12} T^{32} - 91274480 p^{13} T^{33} - 10200029 p^{14} T^{34} + 791392 p^{15} T^{35} + 62614 p^{16} T^{36} - 4624 p^{17} T^{37} - 291 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 - 18 T + 470 T^{2} - 6516 T^{3} + 104201 T^{4} - 1121100 T^{5} + 13431934 T^{6} - 112895598 T^{7} + 1056770664 T^{8} - 6211198182 T^{9} + 42069621398 T^{10} - 40505100180 T^{11} - 508608587109 T^{12} + 19726130422932 T^{13} - 115190196069214 T^{14} + 761930116963614 T^{15} + 6287387668895859 T^{16} - 128665633206649848 T^{17} + 2026059069895604484 T^{18} - 20095962770938094832 T^{19} + \)\(18\!\cdots\!44\)\( T^{20} - 20095962770938094832 p T^{21} + 2026059069895604484 p^{2} T^{22} - 128665633206649848 p^{3} T^{23} + 6287387668895859 p^{4} T^{24} + 761930116963614 p^{5} T^{25} - 115190196069214 p^{6} T^{26} + 19726130422932 p^{7} T^{27} - 508608587109 p^{8} T^{28} - 40505100180 p^{9} T^{29} + 42069621398 p^{10} T^{30} - 6211198182 p^{11} T^{31} + 1056770664 p^{12} T^{32} - 112895598 p^{13} T^{33} + 13431934 p^{14} T^{34} - 1121100 p^{15} T^{35} + 104201 p^{16} T^{36} - 6516 p^{17} T^{37} + 470 p^{18} T^{38} - 18 p^{19} T^{39} + p^{20} T^{40} \)
71 \( 1 + 16 T - 256 T^{2} - 3084 T^{3} + 66425 T^{4} + 512092 T^{5} - 8625144 T^{6} - 36645360 T^{7} + 960694980 T^{8} + 3380563112 T^{9} - 58860816752 T^{10} - 149366684012 T^{11} + 3433021756335 T^{12} + 19056716386500 T^{13} + 18952756972224 T^{14} - 524932892465792 T^{15} - 10473823723399401 T^{16} + 71341917858912984 T^{17} + 2184336262642299176 T^{18} + 1327381202060222744 T^{19} - \)\(14\!\cdots\!00\)\( T^{20} + 1327381202060222744 p T^{21} + 2184336262642299176 p^{2} T^{22} + 71341917858912984 p^{3} T^{23} - 10473823723399401 p^{4} T^{24} - 524932892465792 p^{5} T^{25} + 18952756972224 p^{6} T^{26} + 19056716386500 p^{7} T^{27} + 3433021756335 p^{8} T^{28} - 149366684012 p^{9} T^{29} - 58860816752 p^{10} T^{30} + 3380563112 p^{11} T^{31} + 960694980 p^{12} T^{32} - 36645360 p^{13} T^{33} - 8625144 p^{14} T^{34} + 512092 p^{15} T^{35} + 66425 p^{16} T^{36} - 3084 p^{17} T^{37} - 256 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \)
73 \( 1 - 662 T^{2} + 221179 T^{4} - 49439326 T^{6} + 8334146485 T^{8} - 1138166350304 T^{10} + 132447463647444 T^{12} - 13610776696011280 T^{14} + 1261445493668289578 T^{16} - \)\(10\!\cdots\!72\)\( T^{18} + \)\(81\!\cdots\!98\)\( T^{20} - \)\(10\!\cdots\!72\)\( p^{2} T^{22} + 1261445493668289578 p^{4} T^{24} - 13610776696011280 p^{6} T^{26} + 132447463647444 p^{8} T^{28} - 1138166350304 p^{10} T^{30} + 8334146485 p^{12} T^{32} - 49439326 p^{14} T^{34} + 221179 p^{16} T^{36} - 662 p^{18} T^{38} + p^{20} T^{40} \)
79 \( 1 - 772 T^{2} + 300974 T^{4} - 79250180 T^{6} + 15881528397 T^{8} - 2583290881840 T^{10} + 354721609143528 T^{12} - 42178231844928304 T^{14} + 4416281458244224626 T^{16} - \)\(41\!\cdots\!84\)\( T^{18} + \)\(34\!\cdots\!44\)\( T^{20} - \)\(41\!\cdots\!84\)\( p^{2} T^{22} + 4416281458244224626 p^{4} T^{24} - 42178231844928304 p^{6} T^{26} + 354721609143528 p^{8} T^{28} - 2583290881840 p^{10} T^{30} + 15881528397 p^{12} T^{32} - 79250180 p^{14} T^{34} + 300974 p^{16} T^{36} - 772 p^{18} T^{38} + p^{20} T^{40} \)
83 \( ( 1 - 24 T + 894 T^{2} - 15600 T^{3} + 332865 T^{4} - 4597176 T^{5} + 71783808 T^{6} - 818073720 T^{7} + 10193779926 T^{8} - 97546659000 T^{9} + 1006811916964 T^{10} - 97546659000 p T^{11} + 10193779926 p^{2} T^{12} - 818073720 p^{3} T^{13} + 71783808 p^{4} T^{14} - 4597176 p^{5} T^{15} + 332865 p^{6} T^{16} - 15600 p^{7} T^{17} + 894 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
89 \( 1 + 6 T - 150 T^{2} + 808 T^{3} + 3685 T^{4} - 327028 T^{5} + 1849958 T^{6} + 16865574 T^{7} - 355208492 T^{8} + 3406428486 T^{9} + 23770368054 T^{10} - 554018688884 T^{11} + 2507154486755 T^{12} + 21679381682636 T^{13} - 586967110638826 T^{14} + 35926910654726 p T^{15} + 29570324373338103 T^{16} - 519121278252789240 T^{17} + 2865486495313106852 T^{18} + 22835320674508257412 T^{19} - \)\(51\!\cdots\!48\)\( T^{20} + 22835320674508257412 p T^{21} + 2865486495313106852 p^{2} T^{22} - 519121278252789240 p^{3} T^{23} + 29570324373338103 p^{4} T^{24} + 35926910654726 p^{6} T^{25} - 586967110638826 p^{6} T^{26} + 21679381682636 p^{7} T^{27} + 2507154486755 p^{8} T^{28} - 554018688884 p^{9} T^{29} + 23770368054 p^{10} T^{30} + 3406428486 p^{11} T^{31} - 355208492 p^{12} T^{32} + 16865574 p^{13} T^{33} + 1849958 p^{14} T^{34} - 327028 p^{15} T^{35} + 3685 p^{16} T^{36} + 808 p^{17} T^{37} - 150 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
97 \( 1 - 66 T + 2594 T^{2} - 75372 T^{3} + 1779689 T^{4} - 35813304 T^{5} + 632655514 T^{6} - 10003781694 T^{7} + 143514046128 T^{8} - 1885840605930 T^{9} + 22842952299098 T^{10} - 255964711660032 T^{11} + 2654390580657267 T^{12} - 25383455174125188 T^{13} + 221670781665342902 T^{14} - 1730983594711449630 T^{15} + 11529148182391826187 T^{16} - 57231955975271800200 T^{17} + 81314086133490952836 T^{18} + \)\(24\!\cdots\!92\)\( T^{19} - \)\(35\!\cdots\!60\)\( T^{20} + \)\(24\!\cdots\!92\)\( p T^{21} + 81314086133490952836 p^{2} T^{22} - 57231955975271800200 p^{3} T^{23} + 11529148182391826187 p^{4} T^{24} - 1730983594711449630 p^{5} T^{25} + 221670781665342902 p^{6} T^{26} - 25383455174125188 p^{7} T^{27} + 2654390580657267 p^{8} T^{28} - 255964711660032 p^{9} T^{29} + 22842952299098 p^{10} T^{30} - 1885840605930 p^{11} T^{31} + 143514046128 p^{12} T^{32} - 10003781694 p^{13} T^{33} + 632655514 p^{14} T^{34} - 35813304 p^{15} T^{35} + 1779689 p^{16} T^{36} - 75372 p^{17} T^{37} + 2594 p^{18} T^{38} - 66 p^{19} T^{39} + p^{20} T^{40} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.34366041974543086089987128647, −4.24779354124363893153790700959, −4.12464858090498011693084868742, −4.07923097522969004112733508441, −4.00586583625019691104856269451, −3.99280070110371800721017247755, −3.95745940483377041925926730570, −3.54693171641856956173840258619, −3.51417157787690166020860247077, −3.35407750516282253100758337501, −3.18862487127563121515299737438, −3.16240660010681108494101455996, −3.15299074418077516487611253347, −2.98693751433588238010709913889, −2.90650553003474953498185446163, −2.55562805364076167316249836118, −2.43827826688410862463303771485, −2.28315081403106508849276899203, −2.24766597668038127849594574510, −2.13492692750337816935576321883, −1.95386908057509669650738742730, −1.84899246907087217180698344704, −1.58077064638148416848972069651, −0.825299262928373689453297878164, −0.65019814425119579221552571730, 0.65019814425119579221552571730, 0.825299262928373689453297878164, 1.58077064638148416848972069651, 1.84899246907087217180698344704, 1.95386908057509669650738742730, 2.13492692750337816935576321883, 2.24766597668038127849594574510, 2.28315081403106508849276899203, 2.43827826688410862463303771485, 2.55562805364076167316249836118, 2.90650553003474953498185446163, 2.98693751433588238010709913889, 3.15299074418077516487611253347, 3.16240660010681108494101455996, 3.18862487127563121515299737438, 3.35407750516282253100758337501, 3.51417157787690166020860247077, 3.54693171641856956173840258619, 3.95745940483377041925926730570, 3.99280070110371800721017247755, 4.00586583625019691104856269451, 4.07923097522969004112733508441, 4.12464858090498011693084868742, 4.24779354124363893153790700959, 4.34366041974543086089987128647

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.