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Properties

Label 12-2499e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.436\times 10^{20}$
Sign $1$
Analytic cond. $6.31335\times 10^{7}$
Root an. cond. $4.46705$
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  + 2·2-s + 6·3-s − 4-s + 7·5-s + 12·6-s − 4·8-s + 21·9-s + 14·10-s + 11-s − 6·12-s + 4·13-s + 42·15-s + 6·17-s + 42·18-s + 6·19-s − 7·20-s + 2·22-s + 13·23-s − 24·24-s + 8·25-s + 8·26-s + 56·27-s − 2·29-s + 84·30-s + 16·31-s + 32-s + 6·33-s + ⋯
L(s)  = 1  + 1.41·2-s + 3.46·3-s − 1/2·4-s + 3.13·5-s + 4.89·6-s − 1.41·8-s + 7·9-s + 4.42·10-s + 0.301·11-s − 1.73·12-s + 1.10·13-s + 10.8·15-s + 1.45·17-s + 9.89·18-s + 1.37·19-s − 1.56·20-s + 0.426·22-s + 2.71·23-s − 4.89·24-s + 8/5·25-s + 1.56·26-s + 10.7·27-s − 0.371·29-s + 15.3·30-s + 2.87·31-s + 0.176·32-s + 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{12} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(6.31335\times 10^{7}\)
Root analytic conductor: \(4.46705\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{12} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(321.8284581\)
\(L(\frac12)\) \(\approx\) \(321.8284581\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{6} \)
7 \( 1 \)
17 \( ( 1 - T )^{6} \)
good2 \( 1 - p T + 5 T^{2} - p^{3} T^{3} + 13 T^{4} - 15 T^{5} + 3 p^{3} T^{6} - 15 p T^{7} + 13 p^{2} T^{8} - p^{6} T^{9} + 5 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 7 T + 41 T^{2} - 32 p T^{3} + 558 T^{4} - 1528 T^{5} + 3796 T^{6} - 1528 p T^{7} + 558 p^{2} T^{8} - 32 p^{4} T^{9} + 41 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T + 36 T^{2} - 9 T^{3} + 692 T^{4} - 114 T^{5} + 9302 T^{6} - 114 p T^{7} + 692 p^{2} T^{8} - 9 p^{3} T^{9} + 36 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 4 T + 3 p T^{2} - 11 p T^{3} + 718 T^{4} - 2208 T^{5} + 9873 T^{6} - 2208 p T^{7} + 718 p^{2} T^{8} - 11 p^{4} T^{9} + 3 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 6 T + 74 T^{2} - 238 T^{3} + 1981 T^{4} - 197 p T^{5} + 36944 T^{6} - 197 p^{2} T^{7} + 1981 p^{2} T^{8} - 238 p^{3} T^{9} + 74 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 13 T + 143 T^{2} - 1220 T^{3} + 8542 T^{4} - 50813 T^{5} + 264956 T^{6} - 50813 p T^{7} + 8542 p^{2} T^{8} - 1220 p^{3} T^{9} + 143 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 2 T + 91 T^{2} + 76 T^{3} + 4190 T^{4} + 1643 T^{5} + 140930 T^{6} + 1643 p T^{7} + 4190 p^{2} T^{8} + 76 p^{3} T^{9} + 91 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 16 T + 189 T^{2} - 1403 T^{3} + 9696 T^{4} - 53412 T^{5} + 320307 T^{6} - 53412 p T^{7} + 9696 p^{2} T^{8} - 1403 p^{3} T^{9} + 189 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 5 T + 4 T^{2} - 81 T^{3} + 1252 T^{4} + 3017 T^{5} + 5518 T^{6} + 3017 p T^{7} + 1252 p^{2} T^{8} - 81 p^{3} T^{9} + 4 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 107 T^{2} + 84 T^{3} + 6496 T^{4} - 12 T^{5} + 297606 T^{6} - 12 p T^{7} + 6496 p^{2} T^{8} + 84 p^{3} T^{9} + 107 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 7 T + 154 T^{2} + 14 p T^{3} + 11381 T^{4} + 40800 T^{5} + 627584 T^{6} + 40800 p T^{7} + 11381 p^{2} T^{8} + 14 p^{4} T^{9} + 154 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T + 132 T^{2} - 242 T^{3} + 4851 T^{4} - 1290 T^{5} + 80406 T^{6} - 1290 p T^{7} + 4851 p^{2} T^{8} - 242 p^{3} T^{9} + 132 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 3 T + 194 T^{2} - 161 T^{3} + 358 p T^{4} - 9350 T^{5} + 1262298 T^{6} - 9350 p T^{7} + 358 p^{3} T^{8} - 161 p^{3} T^{9} + 194 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 318 T^{2} - 2754 T^{3} + 42563 T^{4} - 286103 T^{5} + 3229324 T^{6} - 286103 p T^{7} + 42563 p^{2} T^{8} - 2754 p^{3} T^{9} + 318 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 29 T + 594 T^{2} - 8409 T^{3} + 99914 T^{4} - 965436 T^{5} + 8223706 T^{6} - 965436 p T^{7} + 99914 p^{2} T^{8} - 8409 p^{3} T^{9} + 594 p^{4} T^{10} - 29 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T + 251 T^{2} + 1808 T^{3} + 31124 T^{4} + 188128 T^{5} + 2510872 T^{6} + 188128 p T^{7} + 31124 p^{2} T^{8} + 1808 p^{3} T^{9} + 251 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 29 T + 575 T^{2} - 7850 T^{3} + 90604 T^{4} - 868151 T^{5} + 7767584 T^{6} - 868151 p T^{7} + 90604 p^{2} T^{8} - 7850 p^{3} T^{9} + 575 p^{4} T^{10} - 29 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 260 T^{2} - 72 T^{3} + 32617 T^{4} - 38989 T^{5} + 2809870 T^{6} - 38989 p T^{7} + 32617 p^{2} T^{8} - 72 p^{3} T^{9} + 260 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T + 187 T^{2} + 1386 T^{3} + 25200 T^{4} + 157027 T^{5} + 2522632 T^{6} + 157027 p T^{7} + 25200 p^{2} T^{8} + 1386 p^{3} T^{9} + 187 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 29 T + 691 T^{2} - 11342 T^{3} + 161184 T^{4} - 1829498 T^{5} + 18402326 T^{6} - 1829498 p T^{7} + 161184 p^{2} T^{8} - 11342 p^{3} T^{9} + 691 p^{4} T^{10} - 29 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 12 T + 360 T^{2} - 2825 T^{3} + 56560 T^{4} - 331445 T^{5} + 5754882 T^{6} - 331445 p T^{7} + 56560 p^{2} T^{8} - 2825 p^{3} T^{9} + 360 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 366 T^{2} - 2077 T^{3} + 61458 T^{4} - 302786 T^{5} + 6888754 T^{6} - 302786 p T^{7} + 61458 p^{2} T^{8} - 2077 p^{3} T^{9} + 366 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.68222443661536233260083246664, −4.18451092080121341105312507703, −4.14287297805321333040080868807, −4.11149222793424753036286711246, −4.05317580417102372611518005713, −3.89436185582384985748053832781, −3.82590265176299957261838653260, −3.37816559513815546846879128618, −3.37319857235003562108743600178, −3.32097439521797572858645304773, −3.10361848226309338429382374918, −3.09523523694210450662752941363, −2.84590768834833593066925740002, −2.53189058934337504318410608538, −2.42654108327885089397446755324, −2.40155754120542855325629442222, −2.11184831597437598732448699723, −1.92712049937261734403576654307, −1.74179206814114661978674252065, −1.72847232730287405111721710886, −1.52138906751852448108673332812, −1.04259755987670938750380693076, −0.966625887699729556161914479882, −0.892395018859920715081809718466, −0.65782131250893429448052299030, 0.65782131250893429448052299030, 0.892395018859920715081809718466, 0.966625887699729556161914479882, 1.04259755987670938750380693076, 1.52138906751852448108673332812, 1.72847232730287405111721710886, 1.74179206814114661978674252065, 1.92712049937261734403576654307, 2.11184831597437598732448699723, 2.40155754120542855325629442222, 2.42654108327885089397446755324, 2.53189058934337504318410608538, 2.84590768834833593066925740002, 3.09523523694210450662752941363, 3.10361848226309338429382374918, 3.32097439521797572858645304773, 3.37319857235003562108743600178, 3.37816559513815546846879128618, 3.82590265176299957261838653260, 3.89436185582384985748053832781, 4.05317580417102372611518005713, 4.11149222793424753036286711246, 4.14287297805321333040080868807, 4.18451092080121341105312507703, 4.68222443661536233260083246664

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

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