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Properties

Label 12-4014e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.183\times 10^{21}$
Sign $1$
Analytic cond. $1.08424\times 10^{9}$
Root an. cond. $5.66144$
Motivic weight $1$
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  − 6·2-s + 21·4-s − 2·5-s + 8·7-s − 56·8-s + 12·10-s − 2·11-s − 2·13-s − 48·14-s + 126·16-s − 2·17-s − 2·19-s − 42·20-s + 12·22-s − 22·23-s − 7·25-s + 12·26-s + 168·28-s − 8·29-s − 12·31-s − 252·32-s + 12·34-s − 16·35-s + 12·38-s + 112·40-s − 28·41-s + 14·43-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s − 0.894·5-s + 3.02·7-s − 19.7·8-s + 3.79·10-s − 0.603·11-s − 0.554·13-s − 12.8·14-s + 63/2·16-s − 0.485·17-s − 0.458·19-s − 9.39·20-s + 2.55·22-s − 4.58·23-s − 7/5·25-s + 2.35·26-s + 31.7·28-s − 1.48·29-s − 2.15·31-s − 44.5·32-s + 2.05·34-s − 2.70·35-s + 1.94·38-s + 17.7·40-s − 4.37·41-s + 2.13·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 223^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 223^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 223^{6}\)
Sign: $1$
Analytic conductor: \(1.08424\times 10^{9}\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 223^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 + T )^{6} \)
3 \( 1 \)
223 \( ( 1 + T )^{6} \)
good5 \( 1 + 2 T + 11 T^{2} + 4 p T^{3} + 21 p T^{4} + 162 T^{5} + 567 T^{6} + 162 p T^{7} + 21 p^{3} T^{8} + 4 p^{4} T^{9} + 11 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 8 T + 53 T^{2} - 244 T^{3} + 137 p T^{4} - 3142 T^{5} + 8933 T^{6} - 3142 p T^{7} + 137 p^{3} T^{8} - 244 p^{3} T^{9} + 53 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T + 4 p T^{2} + 82 T^{3} + 922 T^{4} + 1454 T^{5} + 12305 T^{6} + 1454 p T^{7} + 922 p^{2} T^{8} + 82 p^{3} T^{9} + 4 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T + 53 T^{2} + 112 T^{3} + 1395 T^{4} + 196 p T^{5} + 22751 T^{6} + 196 p^{2} T^{7} + 1395 p^{2} T^{8} + 112 p^{3} T^{9} + 53 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T + 82 T^{2} + 156 T^{3} + 3084 T^{4} + 5034 T^{5} + 67215 T^{6} + 5034 p T^{7} + 3084 p^{2} T^{8} + 156 p^{3} T^{9} + 82 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T + 79 T^{2} + 82 T^{3} + 2853 T^{4} + 1532 T^{5} + 65377 T^{6} + 1532 p T^{7} + 2853 p^{2} T^{8} + 82 p^{3} T^{9} + 79 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 22 T + 330 T^{2} + 3376 T^{3} + 27572 T^{4} + 177050 T^{5} + 944549 T^{6} + 177050 p T^{7} + 27572 p^{2} T^{8} + 3376 p^{3} T^{9} + 330 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 8 T + 91 T^{2} + 558 T^{3} + 4893 T^{4} + 23934 T^{5} + 161495 T^{6} + 23934 p T^{7} + 4893 p^{2} T^{8} + 558 p^{3} T^{9} + 91 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 12 T + 163 T^{2} + 1446 T^{3} + 11543 T^{4} + 77210 T^{5} + 461997 T^{6} + 77210 p T^{7} + 11543 p^{2} T^{8} + 1446 p^{3} T^{9} + 163 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 146 T^{2} + 16 T^{3} + 9794 T^{4} + 1096 T^{5} + 426551 T^{6} + 1096 p T^{7} + 9794 p^{2} T^{8} + 16 p^{3} T^{9} + 146 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 28 T + 540 T^{2} + 7068 T^{3} + 74874 T^{4} + 627126 T^{5} + 4445591 T^{6} + 627126 p T^{7} + 74874 p^{2} T^{8} + 7068 p^{3} T^{9} + 540 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 14 T + 246 T^{2} - 2256 T^{3} + 23498 T^{4} - 162252 T^{5} + 1270461 T^{6} - 162252 p T^{7} + 23498 p^{2} T^{8} - 2256 p^{3} T^{9} + 246 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 2 T + 161 T^{2} + 258 T^{3} + 11803 T^{4} + 18340 T^{5} + 608617 T^{6} + 18340 p T^{7} + 11803 p^{2} T^{8} + 258 p^{3} T^{9} + 161 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 26 T + 491 T^{2} + 6142 T^{3} + 65029 T^{4} + 553746 T^{5} + 4352587 T^{6} + 553746 p T^{7} + 65029 p^{2} T^{8} + 6142 p^{3} T^{9} + 491 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T + 178 T^{2} - 18 T^{3} + 12718 T^{4} - 53854 T^{5} + 11351 p T^{6} - 53854 p T^{7} + 12718 p^{2} T^{8} - 18 p^{3} T^{9} + 178 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 145 T^{2} + 358 T^{3} + 13345 T^{4} + 44368 T^{5} + 1081027 T^{6} + 44368 p T^{7} + 13345 p^{2} T^{8} + 358 p^{3} T^{9} + 145 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 357 T^{2} - 66 p T^{3} + 52655 T^{4} - 514304 T^{5} + 4532217 T^{6} - 514304 p T^{7} + 52655 p^{2} T^{8} - 66 p^{4} T^{9} + 357 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T + 311 T^{2} + 2688 T^{3} + 43470 T^{4} + 315580 T^{5} + 3864635 T^{6} + 315580 p T^{7} + 43470 p^{2} T^{8} + 2688 p^{3} T^{9} + 311 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 20 T + 432 T^{2} + 5194 T^{3} + 66924 T^{4} + 598592 T^{5} + 5963879 T^{6} + 598592 p T^{7} + 66924 p^{2} T^{8} + 5194 p^{3} T^{9} + 432 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 12 T + 404 T^{2} - 4826 T^{3} + 71522 T^{4} - 769226 T^{5} + 7243841 T^{6} - 769226 p T^{7} + 71522 p^{2} T^{8} - 4826 p^{3} T^{9} + 404 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T + 427 T^{2} + 56 p T^{3} + 81391 T^{4} + 743694 T^{5} + 8769517 T^{6} + 743694 p T^{7} + 81391 p^{2} T^{8} + 56 p^{4} T^{9} + 427 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 2 T + 145 T^{2} + 98 T^{3} - 675 T^{4} + 113172 T^{5} - 1106865 T^{6} + 113172 p T^{7} - 675 p^{2} T^{8} + 98 p^{3} T^{9} + 145 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 6 T + 283 T^{2} + 2374 T^{3} + 46809 T^{4} + 416110 T^{5} + 5208643 T^{6} + 416110 p T^{7} + 46809 p^{2} T^{8} + 2374 p^{3} T^{9} + 283 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.91267651378048146512721125610, −4.50289996410224246416581467565, −4.39915026265323918284280419493, −4.27257979633304088497159559907, −4.21258215017264930753243282187, −4.09955914235445633525735877906, −3.87452469134480289589124007626, −3.73230205945844464290148775015, −3.38047988882223802931563692717, −3.31431895926504089005320084968, −3.31063376797259359189617011724, −3.27415824601307524121922867085, −2.99319709785220111912756857879, −2.51959301591792456052563682626, −2.24092189429598548558871304429, −2.22054872621893301435555082973, −2.21753258108011210017930835239, −2.18659447899924200275657394634, −2.13696249703295312893881942019, −1.63597781482082914912235561339, −1.55483027565923760949399946238, −1.44096500813080645953611542046, −1.35517869892569496348441912434, −1.30350023656048769239704007898, −1.09915659149744250332824598855, 0, 0, 0, 0, 0, 0, 1.09915659149744250332824598855, 1.30350023656048769239704007898, 1.35517869892569496348441912434, 1.44096500813080645953611542046, 1.55483027565923760949399946238, 1.63597781482082914912235561339, 2.13696249703295312893881942019, 2.18659447899924200275657394634, 2.21753258108011210017930835239, 2.22054872621893301435555082973, 2.24092189429598548558871304429, 2.51959301591792456052563682626, 2.99319709785220111912756857879, 3.27415824601307524121922867085, 3.31063376797259359189617011724, 3.31431895926504089005320084968, 3.38047988882223802931563692717, 3.73230205945844464290148775015, 3.87452469134480289589124007626, 4.09955914235445633525735877906, 4.21258215017264930753243282187, 4.27257979633304088497159559907, 4.39915026265323918284280419493, 4.50289996410224246416581467565, 4.91267651378048146512721125610

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

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