Properties

Label 32-61e16-1.1-c1e16-0-0
Degree $32$
Conductor $3.675\times 10^{28}$
Sign $1$
Analytic cond. $1.00394\times 10^{-5}$
Root an. cond. $0.697916$
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  − 5·2-s − 3-s + 10·4-s + 5·6-s + 10·7-s − 10·8-s + 7·9-s − 10·12-s − 12·13-s − 50·14-s + 10·16-s − 35·18-s + 3·19-s − 10·21-s − 15·23-s + 10·24-s + 9·25-s + 60·26-s − 12·27-s + 100·28-s − 15·31-s − 25·32-s + 70·36-s − 5·37-s − 15·38-s + 12·39-s + 12·41-s + ⋯
L(s)  = 1  − 3.53·2-s − 0.577·3-s + 5·4-s + 2.04·6-s + 3.77·7-s − 3.53·8-s + 7/3·9-s − 2.88·12-s − 3.32·13-s − 13.3·14-s + 5/2·16-s − 8.24·18-s + 0.688·19-s − 2.18·21-s − 3.12·23-s + 2.04·24-s + 9/5·25-s + 11.7·26-s − 2.30·27-s + 18.8·28-s − 2.69·31-s − 4.41·32-s + 35/3·36-s − 0.821·37-s − 2.43·38-s + 1.92·39-s + 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(61^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(61^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(61^{16}\)
Sign: $1$
Analytic conductor: \(1.00394\times 10^{-5}\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{61} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 61^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.02173305000\)
\(L(\frac12)\) \(\approx\) \(0.02173305000\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + 53 T + 1264 T^{2} + 17700 T^{3} + 163718 T^{4} + 1199030 T^{5} + 10144001 T^{6} + 106364807 T^{7} + 963246013 T^{8} + 106364807 p T^{9} + 10144001 p^{2} T^{10} + 1199030 p^{3} T^{11} + 163718 p^{4} T^{12} + 17700 p^{5} T^{13} + 1264 p^{6} T^{14} + 53 p^{7} T^{15} + p^{8} T^{16} \)
good2 \( 1 + 5 T + 15 T^{2} + 35 T^{3} + 65 T^{4} + 25 p^{2} T^{5} + 15 p^{3} T^{6} + 55 p T^{7} + 69 T^{8} + 5 T^{9} - 5 p^{3} T^{10} - 85 T^{11} - 85 p T^{12} - 565 T^{13} - 1515 T^{14} - 185 p^{4} T^{15} - 4739 T^{16} - 185 p^{5} T^{17} - 1515 p^{2} T^{18} - 565 p^{3} T^{19} - 85 p^{5} T^{20} - 85 p^{5} T^{21} - 5 p^{9} T^{22} + 5 p^{7} T^{23} + 69 p^{8} T^{24} + 55 p^{10} T^{25} + 15 p^{13} T^{26} + 25 p^{13} T^{27} + 65 p^{12} T^{28} + 35 p^{13} T^{29} + 15 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \)
3 \( 1 + T - 2 p T^{2} - T^{3} + 25 T^{4} - 13 T^{5} - 11 p^{2} T^{6} + 103 T^{7} + 313 T^{8} - 169 p T^{9} - 26 p^{3} T^{10} + 1415 T^{11} + 320 p T^{12} - 3374 T^{13} - 109 T^{14} + 5014 T^{15} - 2597 T^{16} + 5014 p T^{17} - 109 p^{2} T^{18} - 3374 p^{3} T^{19} + 320 p^{5} T^{20} + 1415 p^{5} T^{21} - 26 p^{9} T^{22} - 169 p^{8} T^{23} + 313 p^{8} T^{24} + 103 p^{9} T^{25} - 11 p^{12} T^{26} - 13 p^{11} T^{27} + 25 p^{12} T^{28} - p^{13} T^{29} - 2 p^{15} T^{30} + p^{15} T^{31} + p^{16} T^{32} \)
5 \( 1 - 9 T^{2} + 2 p T^{3} + 57 T^{4} - 44 p T^{5} - 317 T^{6} + 323 p T^{7} + 57 p T^{8} - 2096 p T^{9} + 12537 T^{10} + 11707 p T^{11} - 100648 T^{12} - 7908 p^{2} T^{13} + 748344 T^{14} + 1724 p^{3} T^{15} - 4483684 T^{16} + 1724 p^{4} T^{17} + 748344 p^{2} T^{18} - 7908 p^{5} T^{19} - 100648 p^{4} T^{20} + 11707 p^{6} T^{21} + 12537 p^{6} T^{22} - 2096 p^{8} T^{23} + 57 p^{9} T^{24} + 323 p^{10} T^{25} - 317 p^{10} T^{26} - 44 p^{12} T^{27} + 57 p^{12} T^{28} + 2 p^{14} T^{29} - 9 p^{14} T^{30} + p^{16} T^{32} \)
7 \( 1 - 10 T + 79 T^{2} - 445 T^{3} + 2146 T^{4} - 8705 T^{5} + 31515 T^{6} - 101315 T^{7} + 299913 T^{8} - 117720 p T^{9} + 2202665 T^{10} - 5904770 T^{11} + 16551554 T^{12} - 48186605 T^{13} + 141963314 T^{14} - 406960070 T^{15} + 1109260303 T^{16} - 406960070 p T^{17} + 141963314 p^{2} T^{18} - 48186605 p^{3} T^{19} + 16551554 p^{4} T^{20} - 5904770 p^{5} T^{21} + 2202665 p^{6} T^{22} - 117720 p^{8} T^{23} + 299913 p^{8} T^{24} - 101315 p^{9} T^{25} + 31515 p^{10} T^{26} - 8705 p^{11} T^{27} + 2146 p^{12} T^{28} - 445 p^{13} T^{29} + 79 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \)
11 \( 1 - 89 T^{2} + 3878 T^{4} - 111974 T^{6} + 2446727 T^{8} - 43671170 T^{10} + 666025567 T^{12} - 8872766367 T^{14} + 9454280930 p T^{16} - 8872766367 p^{2} T^{18} + 666025567 p^{4} T^{20} - 43671170 p^{6} T^{22} + 2446727 p^{8} T^{24} - 111974 p^{10} T^{26} + 3878 p^{12} T^{28} - 89 p^{14} T^{30} + p^{16} T^{32} \)
13 \( ( 1 + 6 T + 66 T^{2} + 304 T^{3} + 2198 T^{4} + 8336 T^{5} + 46424 T^{6} + 150617 T^{7} + 707243 T^{8} + 150617 p T^{9} + 46424 p^{2} T^{10} + 8336 p^{3} T^{11} + 2198 p^{4} T^{12} + 304 p^{5} T^{13} + 66 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
17 \( 1 + 43 T^{2} - 295 T^{3} + 1103 T^{4} - 12685 T^{5} + 64103 T^{6} - 315815 T^{7} + 2239621 T^{8} - 9762110 T^{9} + 51703990 T^{10} - 269739230 T^{11} + 1147692367 T^{12} - 5657629095 T^{13} + 24760716085 T^{14} - 102255926960 T^{15} + 460223486907 T^{16} - 102255926960 p T^{17} + 24760716085 p^{2} T^{18} - 5657629095 p^{3} T^{19} + 1147692367 p^{4} T^{20} - 269739230 p^{5} T^{21} + 51703990 p^{6} T^{22} - 9762110 p^{7} T^{23} + 2239621 p^{8} T^{24} - 315815 p^{9} T^{25} + 64103 p^{10} T^{26} - 12685 p^{11} T^{27} + 1103 p^{12} T^{28} - 295 p^{13} T^{29} + 43 p^{14} T^{30} + p^{16} T^{32} \)
19 \( 1 - 3 T - 40 T^{2} + 148 T^{3} + 492 T^{4} - 1623 T^{5} - 142 T^{6} - 1442 p T^{7} - 2364 p T^{8} + 1116180 T^{9} + 2675801 T^{10} - 33259049 T^{11} - 85748771 T^{12} + 851900020 T^{13} + 49181849 p T^{14} - 8222009647 T^{15} - 3932859421 T^{16} - 8222009647 p T^{17} + 49181849 p^{3} T^{18} + 851900020 p^{3} T^{19} - 85748771 p^{4} T^{20} - 33259049 p^{5} T^{21} + 2675801 p^{6} T^{22} + 1116180 p^{7} T^{23} - 2364 p^{9} T^{24} - 1442 p^{10} T^{25} - 142 p^{10} T^{26} - 1623 p^{11} T^{27} + 492 p^{12} T^{28} + 148 p^{13} T^{29} - 40 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \)
23 \( 1 + 15 T + 9 p T^{2} + 1925 T^{3} + 16287 T^{4} + 113670 T^{5} + 722203 T^{6} + 3991415 T^{7} + 19845672 T^{8} + 82824520 T^{9} + 275269985 T^{10} + 370268085 T^{11} - 3381524412 T^{12} - 43228698175 T^{13} - 311471986506 T^{14} - 1860890159125 T^{15} - 9431577763665 T^{16} - 1860890159125 p T^{17} - 311471986506 p^{2} T^{18} - 43228698175 p^{3} T^{19} - 3381524412 p^{4} T^{20} + 370268085 p^{5} T^{21} + 275269985 p^{6} T^{22} + 82824520 p^{7} T^{23} + 19845672 p^{8} T^{24} + 3991415 p^{9} T^{25} + 722203 p^{10} T^{26} + 113670 p^{11} T^{27} + 16287 p^{12} T^{28} + 1925 p^{13} T^{29} + 9 p^{15} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 - 279 T^{2} + 37083 T^{4} - 3118264 T^{6} + 186254757 T^{8} - 8465412925 T^{10} + 310609973592 T^{12} - 9854178730147 T^{14} + 290981028014045 T^{16} - 9854178730147 p^{2} T^{18} + 310609973592 p^{4} T^{20} - 8465412925 p^{6} T^{22} + 186254757 p^{8} T^{24} - 3118264 p^{10} T^{26} + 37083 p^{12} T^{28} - 279 p^{14} T^{30} + p^{16} T^{32} \)
31 \( 1 + 15 T + 229 T^{2} + 2720 T^{3} + 27882 T^{4} + 263495 T^{5} + 2229741 T^{6} + 17793265 T^{7} + 131977326 T^{8} + 925027605 T^{9} + 6198572490 T^{10} + 39477295695 T^{11} + 243905872923 T^{12} + 1454253303145 T^{13} + 8467585214485 T^{14} + 48422784740730 T^{15} + 270849094628077 T^{16} + 48422784740730 p T^{17} + 8467585214485 p^{2} T^{18} + 1454253303145 p^{3} T^{19} + 243905872923 p^{4} T^{20} + 39477295695 p^{5} T^{21} + 6198572490 p^{6} T^{22} + 925027605 p^{7} T^{23} + 131977326 p^{8} T^{24} + 17793265 p^{9} T^{25} + 2229741 p^{10} T^{26} + 263495 p^{11} T^{27} + 27882 p^{12} T^{28} + 2720 p^{13} T^{29} + 229 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \)
37 \( 1 + 5 T + 33 T^{2} - 115 T^{3} - 1393 T^{4} - 9830 T^{5} - 26883 T^{6} + 712835 T^{7} + 3918312 T^{8} + 36125180 T^{9} - 1458715 T^{10} - 373946925 T^{11} - 6588438982 T^{12} - 31199250245 T^{13} + 128367650526 T^{14} + 957957636515 T^{15} + 15506223176675 T^{16} + 957957636515 p T^{17} + 128367650526 p^{2} T^{18} - 31199250245 p^{3} T^{19} - 6588438982 p^{4} T^{20} - 373946925 p^{5} T^{21} - 1458715 p^{6} T^{22} + 36125180 p^{7} T^{23} + 3918312 p^{8} T^{24} + 712835 p^{9} T^{25} - 26883 p^{10} T^{26} - 9830 p^{11} T^{27} - 1393 p^{12} T^{28} - 115 p^{13} T^{29} + 33 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 - 12 T - 5 T^{2} + 443 T^{3} + 2319 T^{4} - 42803 T^{5} + 38281 T^{6} + 1134551 T^{7} + 28007 p T^{8} - 52591508 T^{9} - 2634580 T^{10} + 1723605638 T^{11} - 3821894113 T^{12} + 16168497141 T^{13} - 383721286901 T^{14} + 2026154453570 T^{15} - 6329678380097 T^{16} + 2026154453570 p T^{17} - 383721286901 p^{2} T^{18} + 16168497141 p^{3} T^{19} - 3821894113 p^{4} T^{20} + 1723605638 p^{5} T^{21} - 2634580 p^{6} T^{22} - 52591508 p^{7} T^{23} + 28007 p^{9} T^{24} + 1134551 p^{9} T^{25} + 38281 p^{10} T^{26} - 42803 p^{11} T^{27} + 2319 p^{12} T^{28} + 443 p^{13} T^{29} - 5 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 + 25 T + 475 T^{2} + 5760 T^{3} + 54505 T^{4} + 326750 T^{5} + 794110 T^{6} - 11876585 T^{7} - 96394 p^{2} T^{8} - 1410378900 T^{9} - 5882739295 T^{10} + 2507100035 T^{11} + 269016911220 T^{12} + 2156454943800 T^{13} + 8159908658200 T^{14} - 7528926419265 T^{15} - 230517908176919 T^{16} - 7528926419265 p T^{17} + 8159908658200 p^{2} T^{18} + 2156454943800 p^{3} T^{19} + 269016911220 p^{4} T^{20} + 2507100035 p^{5} T^{21} - 5882739295 p^{6} T^{22} - 1410378900 p^{7} T^{23} - 96394 p^{10} T^{24} - 11876585 p^{9} T^{25} + 794110 p^{10} T^{26} + 326750 p^{11} T^{27} + 54505 p^{12} T^{28} + 5760 p^{13} T^{29} + 475 p^{14} T^{30} + 25 p^{15} T^{31} + p^{16} T^{32} \)
47 \( ( 1 - 3 T + 252 T^{2} - 271 T^{3} + 29293 T^{4} + 1726 T^{5} + 2226609 T^{6} + 1006952 T^{7} + 122448678 T^{8} + 1006952 p T^{9} + 2226609 p^{2} T^{10} + 1726 p^{3} T^{11} + 29293 p^{4} T^{12} - 271 p^{5} T^{13} + 252 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
53 \( 1 + 20 T + 199 T^{2} + 2875 T^{3} + 39397 T^{4} + 370045 T^{5} + 3659993 T^{6} + 38740970 T^{7} + 345763277 T^{8} + 3037043755 T^{9} + 27108033320 T^{10} + 222746060060 T^{11} + 1842983363243 T^{12} + 14984596984095 T^{13} + 111847871710691 T^{14} + 855024909525500 T^{15} + 6543454922608830 T^{16} + 855024909525500 p T^{17} + 111847871710691 p^{2} T^{18} + 14984596984095 p^{3} T^{19} + 1842983363243 p^{4} T^{20} + 222746060060 p^{5} T^{21} + 27108033320 p^{6} T^{22} + 3037043755 p^{7} T^{23} + 345763277 p^{8} T^{24} + 38740970 p^{9} T^{25} + 3659993 p^{10} T^{26} + 370045 p^{11} T^{27} + 39397 p^{12} T^{28} + 2875 p^{13} T^{29} + 199 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \)
59 \( 1 - 5 T + 56 T^{2} + 335 T^{3} + 167 T^{4} + 18855 T^{5} + 14009 T^{6} + 2418195 T^{7} - 245956 p T^{8} + 137830810 T^{9} + 211598925 T^{10} - 447685075 T^{11} + 39793329573 T^{12} + 280053463495 T^{13} + 2697388120680 T^{14} - 3269731262690 T^{15} + 410227757908617 T^{16} - 3269731262690 p T^{17} + 2697388120680 p^{2} T^{18} + 280053463495 p^{3} T^{19} + 39793329573 p^{4} T^{20} - 447685075 p^{5} T^{21} + 211598925 p^{6} T^{22} + 137830810 p^{7} T^{23} - 245956 p^{9} T^{24} + 2418195 p^{9} T^{25} + 14009 p^{10} T^{26} + 18855 p^{11} T^{27} + 167 p^{12} T^{28} + 335 p^{13} T^{29} + 56 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 + 55 T + 1623 T^{2} + 33205 T^{3} + 525513 T^{4} + 6842460 T^{5} + 76504738 T^{6} + 754919110 T^{7} + 6632265321 T^{8} + 50595227995 T^{9} + 304083072180 T^{10} + 881347495735 T^{11} - 10332791560373 T^{12} - 238045875765180 T^{13} - 3030756628702715 T^{14} - 30653009199811070 T^{15} - 267409651963011403 T^{16} - 30653009199811070 p T^{17} - 3030756628702715 p^{2} T^{18} - 238045875765180 p^{3} T^{19} - 10332791560373 p^{4} T^{20} + 881347495735 p^{5} T^{21} + 304083072180 p^{6} T^{22} + 50595227995 p^{7} T^{23} + 6632265321 p^{8} T^{24} + 754919110 p^{9} T^{25} + 76504738 p^{10} T^{26} + 6842460 p^{11} T^{27} + 525513 p^{12} T^{28} + 33205 p^{13} T^{29} + 1623 p^{14} T^{30} + 55 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 50 T + 1310 T^{2} + 25100 T^{3} + 402690 T^{4} + 5710500 T^{5} + 73142420 T^{6} + 857262210 T^{7} + 9291753194 T^{8} + 94226452595 T^{9} + 905415943755 T^{10} + 8315078921580 T^{11} + 73428203558125 T^{12} + 630065735281185 T^{13} + 5326731673336375 T^{14} + 44826640384313440 T^{15} + 377362009506660131 T^{16} + 44826640384313440 p T^{17} + 5326731673336375 p^{2} T^{18} + 630065735281185 p^{3} T^{19} + 73428203558125 p^{4} T^{20} + 8315078921580 p^{5} T^{21} + 905415943755 p^{6} T^{22} + 94226452595 p^{7} T^{23} + 9291753194 p^{8} T^{24} + 857262210 p^{9} T^{25} + 73142420 p^{10} T^{26} + 5710500 p^{11} T^{27} + 402690 p^{12} T^{28} + 25100 p^{13} T^{29} + 1310 p^{14} T^{30} + 50 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 + 11 T - 218 T^{2} - 3889 T^{3} + 6739 T^{4} + 465127 T^{5} + 2249953 T^{6} - 12657591 T^{7} - 198401422 T^{8} - 1622576810 T^{9} - 7165841913 T^{10} + 49748483585 T^{11} + 1050131744229 T^{12} + 13880137357557 T^{13} + 100040852908534 T^{14} - 787338353415830 T^{15} - 17077728803609053 T^{16} - 787338353415830 p T^{17} + 100040852908534 p^{2} T^{18} + 13880137357557 p^{3} T^{19} + 1050131744229 p^{4} T^{20} + 49748483585 p^{5} T^{21} - 7165841913 p^{6} T^{22} - 1622576810 p^{7} T^{23} - 198401422 p^{8} T^{24} - 12657591 p^{9} T^{25} + 2249953 p^{10} T^{26} + 465127 p^{11} T^{27} + 6739 p^{12} T^{28} - 3889 p^{13} T^{29} - 218 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \)
79 \( 1 - 40 T + 1004 T^{2} - 18625 T^{3} + 279368 T^{4} - 3529580 T^{5} + 38355634 T^{6} - 367873670 T^{7} + 3181335467 T^{8} - 26061609010 T^{9} + 212343593095 T^{10} - 22994611830 p T^{11} + 16354414377877 T^{12} - 150158736307295 T^{13} + 1367614192415072 T^{14} - 12190383558121290 T^{15} + 108545585867347675 T^{16} - 12190383558121290 p T^{17} + 1367614192415072 p^{2} T^{18} - 150158736307295 p^{3} T^{19} + 16354414377877 p^{4} T^{20} - 22994611830 p^{6} T^{21} + 212343593095 p^{6} T^{22} - 26061609010 p^{7} T^{23} + 3181335467 p^{8} T^{24} - 367873670 p^{9} T^{25} + 38355634 p^{10} T^{26} - 3529580 p^{11} T^{27} + 279368 p^{12} T^{28} - 18625 p^{13} T^{29} + 1004 p^{14} T^{30} - 40 p^{15} T^{31} + p^{16} T^{32} \)
83 \( 1 - 31 T + 192 T^{2} + 4279 T^{3} - 59466 T^{4} - 263272 T^{5} + 6923583 T^{6} + 27719176 T^{7} - 773399342 T^{8} - 4251313795 T^{9} + 1262640614 p T^{10} + 338765023285 T^{11} - 12069980289866 T^{12} - 6556717409592 T^{13} + 1024908701987209 T^{14} - 197032890173330 T^{15} - 80817926847707993 T^{16} - 197032890173330 p T^{17} + 1024908701987209 p^{2} T^{18} - 6556717409592 p^{3} T^{19} - 12069980289866 p^{4} T^{20} + 338765023285 p^{5} T^{21} + 1262640614 p^{7} T^{22} - 4251313795 p^{7} T^{23} - 773399342 p^{8} T^{24} + 27719176 p^{9} T^{25} + 6923583 p^{10} T^{26} - 263272 p^{11} T^{27} - 59466 p^{12} T^{28} + 4279 p^{13} T^{29} + 192 p^{14} T^{30} - 31 p^{15} T^{31} + p^{16} T^{32} \)
89 \( 1 - 60 T + 1871 T^{2} - 38495 T^{3} + 562467 T^{4} - 5767705 T^{5} + 35049269 T^{6} + 22683470 T^{7} - 3027868689 T^{8} + 27695647285 T^{9} + 59222262600 T^{10} - 4752582400640 T^{11} + 67318776826303 T^{12} - 507535512149835 T^{13} + 865503028135415 T^{14} + 31691773238870840 T^{15} - 459107547574687058 T^{16} + 31691773238870840 p T^{17} + 865503028135415 p^{2} T^{18} - 507535512149835 p^{3} T^{19} + 67318776826303 p^{4} T^{20} - 4752582400640 p^{5} T^{21} + 59222262600 p^{6} T^{22} + 27695647285 p^{7} T^{23} - 3027868689 p^{8} T^{24} + 22683470 p^{9} T^{25} + 35049269 p^{10} T^{26} - 5767705 p^{11} T^{27} + 562467 p^{12} T^{28} - 38495 p^{13} T^{29} + 1871 p^{14} T^{30} - 60 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 - 45 T + 856 T^{2} - 11860 T^{3} + 210304 T^{4} - 3615225 T^{5} + 44038916 T^{6} - 491524840 T^{7} + 6859409486 T^{8} - 85759087530 T^{9} + 818468817393 T^{10} - 8592168079245 T^{11} + 108625344057871 T^{12} - 1098947552679740 T^{13} + 9418474934221653 T^{14} - 104357219141083475 T^{15} + 1183864542942718737 T^{16} - 104357219141083475 p T^{17} + 9418474934221653 p^{2} T^{18} - 1098947552679740 p^{3} T^{19} + 108625344057871 p^{4} T^{20} - 8592168079245 p^{5} T^{21} + 818468817393 p^{6} T^{22} - 85759087530 p^{7} T^{23} + 6859409486 p^{8} T^{24} - 491524840 p^{9} T^{25} + 44038916 p^{10} T^{26} - 3615225 p^{11} T^{27} + 210304 p^{12} T^{28} - 11860 p^{13} T^{29} + 856 p^{14} T^{30} - 45 p^{15} T^{31} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.86965510182611232326668398470, −4.78922503957803231828110830999, −4.71971889267101590411495049390, −4.71207981310758756441980107228, −4.49542464750220258627738104194, −4.48897561443367521807389403789, −4.43229744381759395347850602608, −4.41627503291210573899992858802, −4.35909946469147816004500692360, −3.81841114410427473166483281738, −3.65390006877344087148754636667, −3.62102493617919929483697705711, −3.55115321486051542253146341599, −3.20495135061973258084180686002, −3.18577171183283399854914983262, −3.16094374413185404026970655248, −3.08246231793195354184388249713, −2.72406887281959213704636539487, −2.32617001590758125321994768538, −2.01487242996128374739396393247, −2.01304563296478735404605422620, −1.70640426211790740564416212753, −1.68183952835725434680805745947, −1.63227788354796276569901765105, −1.59992046770254560072212279851, 1.59992046770254560072212279851, 1.63227788354796276569901765105, 1.68183952835725434680805745947, 1.70640426211790740564416212753, 2.01304563296478735404605422620, 2.01487242996128374739396393247, 2.32617001590758125321994768538, 2.72406887281959213704636539487, 3.08246231793195354184388249713, 3.16094374413185404026970655248, 3.18577171183283399854914983262, 3.20495135061973258084180686002, 3.55115321486051542253146341599, 3.62102493617919929483697705711, 3.65390006877344087148754636667, 3.81841114410427473166483281738, 4.35909946469147816004500692360, 4.41627503291210573899992858802, 4.43229744381759395347850602608, 4.48897561443367521807389403789, 4.49542464750220258627738104194, 4.71207981310758756441980107228, 4.71971889267101590411495049390, 4.78922503957803231828110830999, 4.86965510182611232326668398470

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

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