Properties

Label 24-6032e12-1.1-c1e12-0-0
Degree $24$
Conductor $2.320\times 10^{45}$
Sign $1$
Analytic cond. $1.55905\times 10^{20}$
Root an. cond. $6.94015$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $12$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 3·5-s − 6·7-s + 10·9-s − 14·11-s + 12·13-s − 18·15-s + 4·17-s − 11·19-s + 36·21-s − 15·23-s − 23·25-s + 10·27-s + 12·29-s − 13·31-s + 84·33-s − 18·35-s − 23·37-s − 72·39-s − 2·41-s − 26·43-s + 30·45-s − 15·47-s − 16·49-s − 24·51-s + 31·53-s − 42·55-s + ⋯
L(s)  = 1  − 3.46·3-s + 1.34·5-s − 2.26·7-s + 10/3·9-s − 4.22·11-s + 3.32·13-s − 4.64·15-s + 0.970·17-s − 2.52·19-s + 7.85·21-s − 3.12·23-s − 4.59·25-s + 1.92·27-s + 2.22·29-s − 2.33·31-s + 14.6·33-s − 3.04·35-s − 3.78·37-s − 11.5·39-s − 0.312·41-s − 3.96·43-s + 4.47·45-s − 2.18·47-s − 2.28·49-s − 3.36·51-s + 4.25·53-s − 5.66·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 13^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 13^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 13^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(1.55905\times 10^{20}\)
Root analytic conductor: \(6.94015\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(12\)
Selberg data: \((24,\ 2^{48} \cdot 13^{12} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( ( 1 - T )^{12} \)
29 \( ( 1 - T )^{12} \)
good3 \( 1 + 2 p T + 26 T^{2} + 86 T^{3} + 241 T^{4} + 581 T^{5} + 1238 T^{6} + 2330 T^{7} + 1309 p T^{8} + 1978 p T^{9} + 8272 T^{10} + 11437 T^{11} + 17870 T^{12} + 11437 p T^{13} + 8272 p^{2} T^{14} + 1978 p^{4} T^{15} + 1309 p^{5} T^{16} + 2330 p^{5} T^{17} + 1238 p^{6} T^{18} + 581 p^{7} T^{19} + 241 p^{8} T^{20} + 86 p^{9} T^{21} + 26 p^{10} T^{22} + 2 p^{12} T^{23} + p^{12} T^{24} \)
5 \( 1 - 3 T + 32 T^{2} - 87 T^{3} + 526 T^{4} - 1278 T^{5} + 1164 p T^{6} - 12803 T^{7} + 48742 T^{8} - 98149 T^{9} + 327338 T^{10} - 603184 T^{11} + 1803114 T^{12} - 603184 p T^{13} + 327338 p^{2} T^{14} - 98149 p^{3} T^{15} + 48742 p^{4} T^{16} - 12803 p^{5} T^{17} + 1164 p^{7} T^{18} - 1278 p^{7} T^{19} + 526 p^{8} T^{20} - 87 p^{9} T^{21} + 32 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 + 6 T + 52 T^{2} + 242 T^{3} + 1247 T^{4} + 4794 T^{5} + 19294 T^{6} + 64667 T^{7} + 224995 T^{8} + 681278 T^{9} + 2121060 T^{10} + 5850961 T^{11} + 16405230 T^{12} + 5850961 p T^{13} + 2121060 p^{2} T^{14} + 681278 p^{3} T^{15} + 224995 p^{4} T^{16} + 64667 p^{5} T^{17} + 19294 p^{6} T^{18} + 4794 p^{7} T^{19} + 1247 p^{8} T^{20} + 242 p^{9} T^{21} + 52 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 + 14 T + 156 T^{2} + 1266 T^{3} + 9002 T^{4} + 54623 T^{5} + 301640 T^{6} + 1495178 T^{7} + 6872515 T^{8} + 28984312 T^{9} + 114443140 T^{10} + 418790733 T^{11} + 1440658740 T^{12} + 418790733 p T^{13} + 114443140 p^{2} T^{14} + 28984312 p^{3} T^{15} + 6872515 p^{4} T^{16} + 1495178 p^{5} T^{17} + 301640 p^{6} T^{18} + 54623 p^{7} T^{19} + 9002 p^{8} T^{20} + 1266 p^{9} T^{21} + 156 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 4 T + 107 T^{2} - 25 p T^{3} + 5794 T^{4} - 22631 T^{5} + 12497 p T^{6} - 812017 T^{7} + 5920191 T^{8} - 1289521 p T^{9} + 132971100 T^{10} - 465917178 T^{11} + 2474401836 T^{12} - 465917178 p T^{13} + 132971100 p^{2} T^{14} - 1289521 p^{4} T^{15} + 5920191 p^{4} T^{16} - 812017 p^{5} T^{17} + 12497 p^{7} T^{18} - 22631 p^{7} T^{19} + 5794 p^{8} T^{20} - 25 p^{10} T^{21} + 107 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 11 T + 158 T^{2} + 1383 T^{3} + 12295 T^{4} + 87002 T^{5} + 605244 T^{6} + 3620493 T^{7} + 21207087 T^{8} + 110829282 T^{9} + 567324678 T^{10} + 2636941655 T^{11} + 12033589586 T^{12} + 2636941655 p T^{13} + 567324678 p^{2} T^{14} + 110829282 p^{3} T^{15} + 21207087 p^{4} T^{16} + 3620493 p^{5} T^{17} + 605244 p^{6} T^{18} + 87002 p^{7} T^{19} + 12295 p^{8} T^{20} + 1383 p^{9} T^{21} + 158 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 15 T + 11 p T^{2} + 2624 T^{3} + 27300 T^{4} + 223467 T^{5} + 1791735 T^{6} + 12271106 T^{7} + 81873431 T^{8} + 483645371 T^{9} + 2782425964 T^{10} + 14403539115 T^{11} + 72651153000 T^{12} + 14403539115 p T^{13} + 2782425964 p^{2} T^{14} + 483645371 p^{3} T^{15} + 81873431 p^{4} T^{16} + 12271106 p^{5} T^{17} + 1791735 p^{6} T^{18} + 223467 p^{7} T^{19} + 27300 p^{8} T^{20} + 2624 p^{9} T^{21} + 11 p^{11} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 + 13 T + 203 T^{2} + 64 p T^{3} + 21794 T^{4} + 182415 T^{5} + 1603199 T^{6} + 11714250 T^{7} + 88402079 T^{8} + 574888305 T^{9} + 3817224502 T^{10} + 22248268579 T^{11} + 131944454396 T^{12} + 22248268579 p T^{13} + 3817224502 p^{2} T^{14} + 574888305 p^{3} T^{15} + 88402079 p^{4} T^{16} + 11714250 p^{5} T^{17} + 1603199 p^{6} T^{18} + 182415 p^{7} T^{19} + 21794 p^{8} T^{20} + 64 p^{10} T^{21} + 203 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 + 23 T + 527 T^{2} + 7941 T^{3} + 111097 T^{4} + 1276827 T^{5} + 13554258 T^{6} + 127054902 T^{7} + 1103537747 T^{8} + 8728014529 T^{9} + 64166766983 T^{10} + 435446333614 T^{11} + 2751511323734 T^{12} + 435446333614 p T^{13} + 64166766983 p^{2} T^{14} + 8728014529 p^{3} T^{15} + 1103537747 p^{4} T^{16} + 127054902 p^{5} T^{17} + 13554258 p^{6} T^{18} + 1276827 p^{7} T^{19} + 111097 p^{8} T^{20} + 7941 p^{9} T^{21} + 527 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 2 T + 249 T^{2} + 612 T^{3} + 27994 T^{4} + 60878 T^{5} + 1852653 T^{6} + 1490707 T^{7} + 77500173 T^{8} - 183770849 T^{9} + 2149218390 T^{10} - 18788754606 T^{11} + 61043065432 T^{12} - 18788754606 p T^{13} + 2149218390 p^{2} T^{14} - 183770849 p^{3} T^{15} + 77500173 p^{4} T^{16} + 1490707 p^{5} T^{17} + 1852653 p^{6} T^{18} + 60878 p^{7} T^{19} + 27994 p^{8} T^{20} + 612 p^{9} T^{21} + 249 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 26 T + 542 T^{2} + 8290 T^{3} + 110979 T^{4} + 1273931 T^{5} + 13266186 T^{6} + 124801182 T^{7} + 1089007149 T^{8} + 8799532482 T^{9} + 66862958864 T^{10} + 476823178219 T^{11} + 3220721519278 T^{12} + 476823178219 p T^{13} + 66862958864 p^{2} T^{14} + 8799532482 p^{3} T^{15} + 1089007149 p^{4} T^{16} + 124801182 p^{5} T^{17} + 13266186 p^{6} T^{18} + 1273931 p^{7} T^{19} + 110979 p^{8} T^{20} + 8290 p^{9} T^{21} + 542 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 15 T + 472 T^{2} + 5603 T^{3} + 101189 T^{4} + 1010457 T^{5} + 13484528 T^{6} + 116652602 T^{7} + 1261808029 T^{8} + 9589625481 T^{9} + 87962976256 T^{10} + 589967476324 T^{11} + 4701339101226 T^{12} + 589967476324 p T^{13} + 87962976256 p^{2} T^{14} + 9589625481 p^{3} T^{15} + 1261808029 p^{4} T^{16} + 116652602 p^{5} T^{17} + 13484528 p^{6} T^{18} + 1010457 p^{7} T^{19} + 101189 p^{8} T^{20} + 5603 p^{9} T^{21} + 472 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 31 T + 684 T^{2} - 11794 T^{3} + 174667 T^{4} - 2259899 T^{5} + 26435928 T^{6} - 282249014 T^{7} + 2783766107 T^{8} - 25495826095 T^{9} + 218268681532 T^{10} - 1748274710355 T^{11} + 13140250517362 T^{12} - 1748274710355 p T^{13} + 218268681532 p^{2} T^{14} - 25495826095 p^{3} T^{15} + 2783766107 p^{4} T^{16} - 282249014 p^{5} T^{17} + 26435928 p^{6} T^{18} - 2259899 p^{7} T^{19} + 174667 p^{8} T^{20} - 11794 p^{9} T^{21} + 684 p^{10} T^{22} - 31 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 7 T + 292 T^{2} + 2862 T^{3} + 55443 T^{4} + 529763 T^{5} + 7552736 T^{6} + 68653600 T^{7} + 779784039 T^{8} + 6565489441 T^{9} + 64052829788 T^{10} + 490138379137 T^{11} + 4196274957898 T^{12} + 490138379137 p T^{13} + 64052829788 p^{2} T^{14} + 6565489441 p^{3} T^{15} + 779784039 p^{4} T^{16} + 68653600 p^{5} T^{17} + 7552736 p^{6} T^{18} + 529763 p^{7} T^{19} + 55443 p^{8} T^{20} + 2862 p^{9} T^{21} + 292 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 2 T + 383 T^{2} - 1170 T^{3} + 71592 T^{4} - 270018 T^{5} + 8936241 T^{6} - 36892781 T^{7} + 847486809 T^{8} - 3587139597 T^{9} + 65350952156 T^{10} - 271955435964 T^{11} + 4283888677076 T^{12} - 271955435964 p T^{13} + 65350952156 p^{2} T^{14} - 3587139597 p^{3} T^{15} + 847486809 p^{4} T^{16} - 36892781 p^{5} T^{17} + 8936241 p^{6} T^{18} - 270018 p^{7} T^{19} + 71592 p^{8} T^{20} - 1170 p^{9} T^{21} + 383 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 47 T + 1391 T^{2} + 29102 T^{3} + 489824 T^{4} + 6842417 T^{5} + 83611111 T^{6} + 911682012 T^{7} + 9201726495 T^{8} + 87074101065 T^{9} + 789755296322 T^{10} + 6851131247955 T^{11} + 57329291308128 T^{12} + 6851131247955 p T^{13} + 789755296322 p^{2} T^{14} + 87074101065 p^{3} T^{15} + 9201726495 p^{4} T^{16} + 911682012 p^{5} T^{17} + 83611111 p^{6} T^{18} + 6842417 p^{7} T^{19} + 489824 p^{8} T^{20} + 29102 p^{9} T^{21} + 1391 p^{10} T^{22} + 47 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 32 T + 864 T^{2} + 15182 T^{3} + 242911 T^{4} + 43369 p T^{5} + 37319970 T^{6} + 383901588 T^{7} + 3911271189 T^{8} + 34905667906 T^{9} + 319866412006 T^{10} + 2638091724135 T^{11} + 23230088777414 T^{12} + 2638091724135 p T^{13} + 319866412006 p^{2} T^{14} + 34905667906 p^{3} T^{15} + 3911271189 p^{4} T^{16} + 383901588 p^{5} T^{17} + 37319970 p^{6} T^{18} + 43369 p^{8} T^{19} + 242911 p^{8} T^{20} + 15182 p^{9} T^{21} + 864 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 + 25 T + 701 T^{2} + 142 p T^{3} + 166109 T^{4} + 1715108 T^{5} + 20488920 T^{6} + 159408272 T^{7} + 1642150173 T^{8} + 9783501343 T^{9} + 1368681663 p T^{10} + 474031422006 T^{11} + 6188129878514 T^{12} + 474031422006 p T^{13} + 1368681663 p^{3} T^{14} + 9783501343 p^{3} T^{15} + 1642150173 p^{4} T^{16} + 159408272 p^{5} T^{17} + 20488920 p^{6} T^{18} + 1715108 p^{7} T^{19} + 166109 p^{8} T^{20} + 142 p^{10} T^{21} + 701 p^{10} T^{22} + 25 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 + 7 T + 644 T^{2} + 5011 T^{3} + 207841 T^{4} + 1658026 T^{5} + 44047894 T^{6} + 341014101 T^{7} + 6759001741 T^{8} + 48859064606 T^{9} + 784813688954 T^{10} + 5141250637431 T^{11} + 70356045166394 T^{12} + 5141250637431 p T^{13} + 784813688954 p^{2} T^{14} + 48859064606 p^{3} T^{15} + 6759001741 p^{4} T^{16} + 341014101 p^{5} T^{17} + 44047894 p^{6} T^{18} + 1658026 p^{7} T^{19} + 207841 p^{8} T^{20} + 5011 p^{9} T^{21} + 644 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 12 T + 377 T^{2} + 2501 T^{3} + 68961 T^{4} + 351158 T^{5} + 10399159 T^{6} + 41900937 T^{7} + 1249478527 T^{8} + 3932227465 T^{9} + 128853255624 T^{10} + 368689262585 T^{11} + 11656756646334 T^{12} + 368689262585 p T^{13} + 128853255624 p^{2} T^{14} + 3932227465 p^{3} T^{15} + 1249478527 p^{4} T^{16} + 41900937 p^{5} T^{17} + 10399159 p^{6} T^{18} + 351158 p^{7} T^{19} + 68961 p^{8} T^{20} + 2501 p^{9} T^{21} + 377 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 6 T + 618 T^{2} - 3369 T^{3} + 199185 T^{4} - 999397 T^{5} + 43292554 T^{6} - 199625645 T^{7} + 6984541713 T^{8} - 29447730255 T^{9} + 875559773236 T^{10} - 3345047976832 T^{11} + 87162624346730 T^{12} - 3345047976832 p T^{13} + 875559773236 p^{2} T^{14} - 29447730255 p^{3} T^{15} + 6984541713 p^{4} T^{16} - 199625645 p^{5} T^{17} + 43292554 p^{6} T^{18} - 999397 p^{7} T^{19} + 199185 p^{8} T^{20} - 3369 p^{9} T^{21} + 618 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 7 T + 522 T^{2} + 2985 T^{3} + 130297 T^{4} + 549974 T^{5} + 22021088 T^{6} + 66883255 T^{7} + 3066114053 T^{8} + 7948793268 T^{9} + 379548140958 T^{10} + 965025983599 T^{11} + 40358799388146 T^{12} + 965025983599 p T^{13} + 379548140958 p^{2} T^{14} + 7948793268 p^{3} T^{15} + 3066114053 p^{4} T^{16} + 66883255 p^{5} T^{17} + 22021088 p^{6} T^{18} + 549974 p^{7} T^{19} + 130297 p^{8} T^{20} + 2985 p^{9} T^{21} + 522 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.91841558039042149134097175623, −2.77520174075051665223503034893, −2.59792359795614183816045409658, −2.54299497303602635817328152595, −2.42411607273807744975796963617, −2.31744108971742468701819422231, −2.28658837788283797201483568829, −2.28314095469861550049155237885, −2.26763996182315928761412599101, −2.18789037806176270964383568874, −2.09975017712733764407909715059, −2.03730183275126946361421262214, −1.93068211209253327173950474566, −1.83335914406374925960547886706, −1.57618769335834469782539320356, −1.51016737819915345317252164707, −1.49540225493514714066437307352, −1.47404976925206405309265470462, −1.42075200984278426150844656477, −1.28813824918565910638809332666, −1.18836396000345977062397913022, −1.16143971979236232990214745930, −1.10496057446964135483530556495, −0.945296696636214778323877324570, −0.73527112220342430680065062182, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.73527112220342430680065062182, 0.945296696636214778323877324570, 1.10496057446964135483530556495, 1.16143971979236232990214745930, 1.18836396000345977062397913022, 1.28813824918565910638809332666, 1.42075200984278426150844656477, 1.47404976925206405309265470462, 1.49540225493514714066437307352, 1.51016737819915345317252164707, 1.57618769335834469782539320356, 1.83335914406374925960547886706, 1.93068211209253327173950474566, 2.03730183275126946361421262214, 2.09975017712733764407909715059, 2.18789037806176270964383568874, 2.26763996182315928761412599101, 2.28314095469861550049155237885, 2.28658837788283797201483568829, 2.31744108971742468701819422231, 2.42411607273807744975796963617, 2.54299497303602635817328152595, 2.59792359795614183816045409658, 2.77520174075051665223503034893, 2.91841558039042149134097175623

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

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