Properties

Label 12-245e6-1.1-c5e6-0-0
Degree $12$
Conductor $2.163\times 10^{14}$
Sign $1$
Analytic cond. $3.68094\times 10^{9}$
Root an. cond. $6.26849$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 20·3-s − 68·4-s − 150·5-s − 100·6-s + 390·8-s − 340·9-s + 750·10-s − 924·11-s − 1.36e3·12-s − 150·13-s − 3.00e3·15-s + 2.07e3·16-s + 1.54e3·17-s + 1.70e3·18-s + 92·19-s + 1.02e4·20-s + 4.62e3·22-s − 3.92e3·23-s + 7.80e3·24-s + 1.31e4·25-s + 750·26-s − 1.04e4·27-s + 1.26e3·29-s + 1.50e4·30-s − 7.16e3·31-s − 1.17e4·32-s + ⋯
L(s)  = 1  − 0.883·2-s + 1.28·3-s − 2.12·4-s − 2.68·5-s − 1.13·6-s + 2.15·8-s − 1.39·9-s + 2.37·10-s − 2.30·11-s − 2.72·12-s − 0.246·13-s − 3.44·15-s + 2.03·16-s + 1.29·17-s + 1.23·18-s + 0.0584·19-s + 5.70·20-s + 2.03·22-s − 1.54·23-s + 2.76·24-s + 21/5·25-s + 0.217·26-s − 2.74·27-s + 0.279·29-s + 3.04·30-s − 1.33·31-s − 2.02·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(3.68094\times 10^{9}\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{245} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 5^{6} \cdot 7^{12} ,\ ( \ : [5/2]^{6} ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + p^{2} T )^{6} \)
7 \( 1 \)
good2 \( 1 + 5 T + 93 T^{2} + 415 T^{3} + 2185 p T^{4} + 3785 p^{2} T^{5} + 19281 p^{3} T^{6} + 3785 p^{7} T^{7} + 2185 p^{11} T^{8} + 415 p^{15} T^{9} + 93 p^{20} T^{10} + 5 p^{25} T^{11} + p^{30} T^{12} \)
3 \( 1 - 20 T + 740 T^{2} - 11200 T^{3} + 30640 p^{2} T^{4} - 482020 p^{2} T^{5} + 3044590 p^{3} T^{6} - 482020 p^{7} T^{7} + 30640 p^{12} T^{8} - 11200 p^{15} T^{9} + 740 p^{20} T^{10} - 20 p^{25} T^{11} + p^{30} T^{12} \)
11 \( 1 + 84 p T + 825717 T^{2} + 447883644 T^{3} + 251513820115 T^{4} + 108565023355632 T^{5} + 48862291165415198 T^{6} + 108565023355632 p^{5} T^{7} + 251513820115 p^{10} T^{8} + 447883644 p^{15} T^{9} + 825717 p^{20} T^{10} + 84 p^{26} T^{11} + p^{30} T^{12} \)
13 \( 1 + 150 T + 796287 T^{2} + 297858110 T^{3} + 464061438907 T^{4} + 133413930219380 T^{5} + 231222100617263226 T^{6} + 133413930219380 p^{5} T^{7} + 464061438907 p^{10} T^{8} + 297858110 p^{15} T^{9} + 796287 p^{20} T^{10} + 150 p^{25} T^{11} + p^{30} T^{12} \)
17 \( 1 - 1540 T + 7451530 T^{2} - 8871686260 T^{3} + 24111152130175 T^{4} - 22932939816405640 T^{5} + 44337762427137456300 T^{6} - 22932939816405640 p^{5} T^{7} + 24111152130175 p^{10} T^{8} - 8871686260 p^{15} T^{9} + 7451530 p^{20} T^{10} - 1540 p^{25} T^{11} + p^{30} T^{12} \)
19 \( 1 - 92 T + 7512461 T^{2} - 3685957332 T^{3} + 25688116768307 T^{4} - 23303405428560392 T^{5} + 63936283449584109166 T^{6} - 23303405428560392 p^{5} T^{7} + 25688116768307 p^{10} T^{8} - 3685957332 p^{15} T^{9} + 7512461 p^{20} T^{10} - 92 p^{25} T^{11} + p^{30} T^{12} \)
23 \( 1 + 3920 T + 24036187 T^{2} + 81532050560 T^{3} + 328692879009670 T^{4} + 886267605267526160 T^{5} + \)\(25\!\cdots\!07\)\( T^{6} + 886267605267526160 p^{5} T^{7} + 328692879009670 p^{10} T^{8} + 81532050560 p^{15} T^{9} + 24036187 p^{20} T^{10} + 3920 p^{25} T^{11} + p^{30} T^{12} \)
29 \( 1 - 1264 T + 34338048 T^{2} - 97759319540 T^{3} + 1265693527774624 T^{4} - 2479979403250472560 T^{5} + \)\(30\!\cdots\!54\)\( T^{6} - 2479979403250472560 p^{5} T^{7} + 1265693527774624 p^{10} T^{8} - 97759319540 p^{15} T^{9} + 34338048 p^{20} T^{10} - 1264 p^{25} T^{11} + p^{30} T^{12} \)
31 \( 1 + 7160 T + 91720490 T^{2} + 813875493096 T^{3} + 5583909231338895 T^{4} + 36208570257510880880 T^{5} + \)\(69\!\cdots\!76\)\( p T^{6} + 36208570257510880880 p^{5} T^{7} + 5583909231338895 p^{10} T^{8} + 813875493096 p^{15} T^{9} + 91720490 p^{20} T^{10} + 7160 p^{25} T^{11} + p^{30} T^{12} \)
37 \( 1 + 14170 T + 318813415 T^{2} + 3192181392650 T^{3} + 46176184376023275 T^{4} + \)\(36\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!50\)\( T^{6} + \)\(36\!\cdots\!20\)\( p^{5} T^{7} + 46176184376023275 p^{10} T^{8} + 3192181392650 p^{15} T^{9} + 318813415 p^{20} T^{10} + 14170 p^{25} T^{11} + p^{30} T^{12} \)
41 \( 1 - 4098 T + 461440565 T^{2} - 793603240022 T^{3} + 101430559965472602 T^{4} - 78439540555050663786 T^{5} + \)\(14\!\cdots\!01\)\( T^{6} - 78439540555050663786 p^{5} T^{7} + 101430559965472602 p^{10} T^{8} - 793603240022 p^{15} T^{9} + 461440565 p^{20} T^{10} - 4098 p^{25} T^{11} + p^{30} T^{12} \)
43 \( 1 + 24460 T + 767727092 T^{2} + 12925607212560 T^{3} + 244456305714648992 T^{4} + \)\(31\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!66\)\( T^{6} + \)\(31\!\cdots\!20\)\( p^{5} T^{7} + 244456305714648992 p^{10} T^{8} + 12925607212560 p^{15} T^{9} + 767727092 p^{20} T^{10} + 24460 p^{25} T^{11} + p^{30} T^{12} \)
47 \( 1 - 42940 T + 1662499309 T^{2} - 39067396764460 T^{3} + 894007089501510835 T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!94\)\( T^{6} - \)\(15\!\cdots\!00\)\( p^{5} T^{7} + 894007089501510835 p^{10} T^{8} - 39067396764460 p^{15} T^{9} + 1662499309 p^{20} T^{10} - 42940 p^{25} T^{11} + p^{30} T^{12} \)
53 \( 1 + 2450 T + 1731298775 T^{2} + 12557810325090 T^{3} + 1332901811960979875 T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(65\!\cdots\!30\)\( T^{6} + \)\(13\!\cdots\!00\)\( p^{5} T^{7} + 1332901811960979875 p^{10} T^{8} + 12557810325090 p^{15} T^{9} + 1731298775 p^{20} T^{10} + 2450 p^{25} T^{11} + p^{30} T^{12} \)
59 \( 1 + 64600 T + 4003051370 T^{2} + 159881299023080 T^{3} + 6047720347419053255 T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!48\)\( T^{6} + \)\(18\!\cdots\!00\)\( p^{5} T^{7} + 6047720347419053255 p^{10} T^{8} + 159881299023080 p^{15} T^{9} + 4003051370 p^{20} T^{10} + 64600 p^{25} T^{11} + p^{30} T^{12} \)
61 \( 1 - 73620 T + 5595394780 T^{2} - 262435447105344 T^{3} + 11885303041254933320 T^{4} - \)\(40\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!86\)\( T^{6} - \)\(40\!\cdots\!80\)\( p^{5} T^{7} + 11885303041254933320 p^{10} T^{8} - 262435447105344 p^{15} T^{9} + 5595394780 p^{20} T^{10} - 73620 p^{25} T^{11} + p^{30} T^{12} \)
67 \( 1 + 142620 T + 10633871464 T^{2} + 485585751900360 T^{3} + 13732683630728961880 T^{4} + \)\(19\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!34\)\( T^{6} + \)\(19\!\cdots\!40\)\( p^{5} T^{7} + 13732683630728961880 p^{10} T^{8} + 485585751900360 p^{15} T^{9} + 10633871464 p^{20} T^{10} + 142620 p^{25} T^{11} + p^{30} T^{12} \)
71 \( 1 + 154256 T + 15903461602 T^{2} + 1239493386418352 T^{3} + 78922551198445875343 T^{4} + \)\(42\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(42\!\cdots\!92\)\( p^{5} T^{7} + 78922551198445875343 p^{10} T^{8} + 1239493386418352 p^{15} T^{9} + 15903461602 p^{20} T^{10} + 154256 p^{25} T^{11} + p^{30} T^{12} \)
73 \( 1 + 5120 T + 6905477830 T^{2} + 111041937751360 T^{3} + 25601871633036081215 T^{4} + \)\(45\!\cdots\!80\)\( T^{5} + \)\(64\!\cdots\!60\)\( T^{6} + \)\(45\!\cdots\!80\)\( p^{5} T^{7} + 25601871633036081215 p^{10} T^{8} + 111041937751360 p^{15} T^{9} + 6905477830 p^{20} T^{10} + 5120 p^{25} T^{11} + p^{30} T^{12} \)
79 \( 1 + 222504 T + 30790489134 T^{2} + 3146130332337464 T^{3} + \)\(26\!\cdots\!03\)\( T^{4} + \)\(18\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!04\)\( p^{5} T^{7} + \)\(26\!\cdots\!03\)\( p^{10} T^{8} + 3146130332337464 p^{15} T^{9} + 30790489134 p^{20} T^{10} + 222504 p^{25} T^{11} + p^{30} T^{12} \)
83 \( 1 - 179580 T + 26415024920 T^{2} - 2643470654451640 T^{3} + \)\(24\!\cdots\!72\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} - \)\(18\!\cdots\!40\)\( p^{5} T^{7} + \)\(24\!\cdots\!72\)\( p^{10} T^{8} - 2643470654451640 p^{15} T^{9} + 26415024920 p^{20} T^{10} - 179580 p^{25} T^{11} + p^{30} T^{12} \)
89 \( 1 - 41648 T + 14985273296 T^{2} - 711975402452908 T^{3} + \)\(13\!\cdots\!52\)\( T^{4} - \)\(54\!\cdots\!48\)\( T^{5} + \)\(89\!\cdots\!46\)\( T^{6} - \)\(54\!\cdots\!48\)\( p^{5} T^{7} + \)\(13\!\cdots\!52\)\( p^{10} T^{8} - 711975402452908 p^{15} T^{9} + 14985273296 p^{20} T^{10} - 41648 p^{25} T^{11} + p^{30} T^{12} \)
97 \( 1 + 73980 T + 24090535458 T^{2} + 2757215884896140 T^{3} + \)\(39\!\cdots\!07\)\( T^{4} + \)\(34\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!84\)\( T^{6} + \)\(34\!\cdots\!00\)\( p^{5} T^{7} + \)\(39\!\cdots\!07\)\( p^{10} T^{8} + 2757215884896140 p^{15} T^{9} + 24090535458 p^{20} T^{10} + 73980 p^{25} T^{11} + p^{30} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.27477490502836555546470677380, −6.02737344276277312033647336075, −5.55496298657086659619928415561, −5.52312059150619166021419666986, −5.51552553048396383021758408746, −5.43995499433310503026776973112, −5.12510116462414021518067263054, −4.76439962533707293214923117743, −4.64021937632334924522700308637, −4.42138424010970918297167830168, −4.22747393210658824808671607627, −4.06318157292085137732634478981, −4.02608668632801916178858915846, −3.48053194199081156568177165902, −3.44864088181363275860441974354, −3.21873185990455479347471450994, −3.16387213929912410247230503341, −2.77393816521697384147327131406, −2.70189779788680767413977195466, −2.42132571445461832794180889249, −2.23991617151723724419948030948, −1.53698927422100383023646048256, −1.43751445572590652987713545016, −1.05822187658478752204260619945, −0.936751823763627328388798543442, 0, 0, 0, 0, 0, 0, 0.936751823763627328388798543442, 1.05822187658478752204260619945, 1.43751445572590652987713545016, 1.53698927422100383023646048256, 2.23991617151723724419948030948, 2.42132571445461832794180889249, 2.70189779788680767413977195466, 2.77393816521697384147327131406, 3.16387213929912410247230503341, 3.21873185990455479347471450994, 3.44864088181363275860441974354, 3.48053194199081156568177165902, 4.02608668632801916178858915846, 4.06318157292085137732634478981, 4.22747393210658824808671607627, 4.42138424010970918297167830168, 4.64021937632334924522700308637, 4.76439962533707293214923117743, 5.12510116462414021518067263054, 5.43995499433310503026776973112, 5.51552553048396383021758408746, 5.52312059150619166021419666986, 5.55496298657086659619928415561, 6.02737344276277312033647336075, 6.27477490502836555546470677380

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

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