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Properties

Label 12-5415e6-1.1-c1e6-0-2
Degree $12$
Conductor $2.521\times 10^{22}$
Sign $1$
Analytic cond. $6.53511\times 10^{9}$
Root an. cond. $6.57563$
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  + 3·2-s + 6·3-s + 6·5-s + 18·6-s + 4·7-s − 9·8-s + 21·9-s + 18·10-s + 8·13-s + 12·14-s + 36·15-s − 9·16-s + 3·17-s + 63·18-s + 24·21-s + 3·23-s − 54·24-s + 21·25-s + 24·26-s + 56·27-s + 5·29-s + 108·30-s + 11·31-s + 32-s + 9·34-s + 24·35-s − 7·37-s + ⋯
L(s)  = 1  + 2.12·2-s + 3.46·3-s + 2.68·5-s + 7.34·6-s + 1.51·7-s − 3.18·8-s + 7·9-s + 5.69·10-s + 2.21·13-s + 3.20·14-s + 9.29·15-s − 9/4·16-s + 0.727·17-s + 14.8·18-s + 5.23·21-s + 0.625·23-s − 11.0·24-s + 21/5·25-s + 4.70·26-s + 10.7·27-s + 0.928·29-s + 19.7·30-s + 1.97·31-s + 0.176·32-s + 1.54·34-s + 4.05·35-s − 1.15·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(6.53511\times 10^{9}\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(870.8862117\)
\(L(\frac12)\) \(\approx\) \(870.8862117\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{6} \)
5 \( ( 1 - T )^{6} \)
19 \( 1 \)
good2 \( 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 55 T^{5} + 87 T^{6} - 55 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 4 T + 38 T^{2} - 121 T^{3} + 613 T^{4} - 1563 T^{5} + 5545 T^{6} - 1563 p T^{7} + 613 p^{2} T^{8} - 121 p^{3} T^{9} + 38 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 p T^{2} - 21 T^{3} + 549 T^{4} - 544 T^{5} + 6783 T^{6} - 544 p T^{7} + 549 p^{2} T^{8} - 21 p^{3} T^{9} + 3 p^{5} T^{10} + p^{6} T^{12} \)
13 \( 1 - 8 T + 79 T^{2} - 432 T^{3} + 2510 T^{4} - 10168 T^{5} + 42963 T^{6} - 10168 p T^{7} + 2510 p^{2} T^{8} - 432 p^{3} T^{9} + 79 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T + 58 T^{2} - 164 T^{3} + 1754 T^{4} - 4039 T^{5} + 35517 T^{6} - 4039 p T^{7} + 1754 p^{2} T^{8} - 164 p^{3} T^{9} + 58 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 3 T + 76 T^{2} - 164 T^{3} + 2856 T^{4} - 4207 T^{5} + 75541 T^{6} - 4207 p T^{7} + 2856 p^{2} T^{8} - 164 p^{3} T^{9} + 76 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 167 T^{2} - 23 p T^{3} + 11764 T^{4} - 37102 T^{5} + 450297 T^{6} - 37102 p T^{7} + 11764 p^{2} T^{8} - 23 p^{4} T^{9} + 167 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 11 T + 142 T^{2} - 1136 T^{3} + 8410 T^{4} - 54163 T^{5} + 309333 T^{6} - 54163 p T^{7} + 8410 p^{2} T^{8} - 1136 p^{3} T^{9} + 142 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 7 T + 212 T^{2} + 1111 T^{3} + 18541 T^{4} + 74860 T^{5} + 892685 T^{6} + 74860 p T^{7} + 18541 p^{2} T^{8} + 1111 p^{3} T^{9} + 212 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 25 T + 468 T^{2} - 5969 T^{3} + 62899 T^{4} - 524566 T^{5} + 3722073 T^{6} - 524566 p T^{7} + 62899 p^{2} T^{8} - 5969 p^{3} T^{9} + 468 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T + 3 p T^{2} + 378 T^{3} + 9525 T^{4} + 28222 T^{5} + 481803 T^{6} + 28222 p T^{7} + 9525 p^{2} T^{8} + 378 p^{3} T^{9} + 3 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 12 T + 222 T^{2} - 1792 T^{3} + 21377 T^{4} - 139853 T^{5} + 1273789 T^{6} - 139853 p T^{7} + 21377 p^{2} T^{8} - 1792 p^{3} T^{9} + 222 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 222 T^{2} - 1604 T^{3} + 12036 T^{4} - 43936 T^{5} + 324753 T^{6} - 43936 p T^{7} + 12036 p^{2} T^{8} - 1604 p^{3} T^{9} + 222 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T + 147 T^{2} - 539 T^{3} + 11798 T^{4} - 29194 T^{5} + 771337 T^{6} - 29194 p T^{7} + 11798 p^{2} T^{8} - 539 p^{3} T^{9} + 147 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T + 185 T^{2} - 884 T^{3} + 19014 T^{4} - 80907 T^{5} + 1396439 T^{6} - 80907 p T^{7} + 19014 p^{2} T^{8} - 884 p^{3} T^{9} + 185 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T + 147 T^{2} + 739 T^{3} + 17746 T^{4} + 76290 T^{5} + 1352555 T^{6} + 76290 p T^{7} + 17746 p^{2} T^{8} + 739 p^{3} T^{9} + 147 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 7 T + 234 T^{2} - 392 T^{3} + 21283 T^{4} + 29144 T^{5} + 1558875 T^{6} + 29144 p T^{7} + 21283 p^{2} T^{8} - 392 p^{3} T^{9} + 234 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 8 T + 129 T^{2} + 1530 T^{3} + 15252 T^{4} + 111785 T^{5} + 1372969 T^{6} + 111785 p T^{7} + 15252 p^{2} T^{8} + 1530 p^{3} T^{9} + 129 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 49 T + 1360 T^{2} - 26551 T^{3} + 399341 T^{4} - 4800874 T^{5} + 47121677 T^{6} - 4800874 p T^{7} + 399341 p^{2} T^{8} - 26551 p^{3} T^{9} + 1360 p^{4} T^{10} - 49 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 19 T + 274 T^{2} - 2609 T^{3} + 30653 T^{4} - 283376 T^{5} + 2944995 T^{6} - 283376 p T^{7} + 30653 p^{2} T^{8} - 2609 p^{3} T^{9} + 274 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T - 123 T^{2} - 319 T^{3} + 22438 T^{4} - 24384 T^{5} - 1453773 T^{6} - 24384 p T^{7} + 22438 p^{2} T^{8} - 319 p^{3} T^{9} - 123 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 15 T + 486 T^{2} - 5640 T^{3} + 101326 T^{4} - 946135 T^{5} + 12333797 T^{6} - 946135 p T^{7} + 101326 p^{2} T^{8} - 5640 p^{3} T^{9} + 486 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.22640132434898762371999752364, −3.91777965120637257482195825128, −3.79712855020410297606856408868, −3.75531046716474622972750827231, −3.69840216930407171157963805868, −3.68113763107850318893555061287, −3.44304492062120786842881168955, −3.15143103850553151602908004290, −3.00884954085218462086797127845, −2.92398589679201441670191032658, −2.89729835208218619078087277267, −2.73581567576294276546331722513, −2.44875929393673051412492030175, −2.36474916172626863919958116448, −2.25692831072358000680934021416, −2.01342276698971058369785919511, −1.83196004006875826781527536047, −1.83097614984092812239379593065, −1.67125408184862315304646532154, −1.41774388043875070844852832415, −1.26104936928853866168046391338, −0.921917384610589827589490678274, −0.795331247348206403512868244345, −0.71985303984409387555518026174, −0.69746666135231138126579941289, 0.69746666135231138126579941289, 0.71985303984409387555518026174, 0.795331247348206403512868244345, 0.921917384610589827589490678274, 1.26104936928853866168046391338, 1.41774388043875070844852832415, 1.67125408184862315304646532154, 1.83097614984092812239379593065, 1.83196004006875826781527536047, 2.01342276698971058369785919511, 2.25692831072358000680934021416, 2.36474916172626863919958116448, 2.44875929393673051412492030175, 2.73581567576294276546331722513, 2.89729835208218619078087277267, 2.92398589679201441670191032658, 3.00884954085218462086797127845, 3.15143103850553151602908004290, 3.44304492062120786842881168955, 3.68113763107850318893555061287, 3.69840216930407171157963805868, 3.75531046716474622972750827231, 3.79712855020410297606856408868, 3.91777965120637257482195825128, 4.22640132434898762371999752364

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

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