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Properties

Label 12-57e12-1.1-c1e6-0-0
Degree $12$
Conductor $1.176\times 10^{21}$
Sign $1$
Analytic cond. $3.04902\times 10^{8}$
Root an. cond. $5.09346$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 9·5-s + 9·7-s + 8-s − 9·11-s + 3·13-s + 3·16-s − 15·17-s + 27·20-s − 6·23-s + 33·25-s − 27·28-s + 15·29-s − 81·35-s − 6·37-s − 9·40-s − 6·41-s + 15·43-s + 27·44-s − 9·47-s + 24·49-s − 9·52-s + 6·53-s + 81·55-s + 9·56-s + 15·59-s + 3·61-s + ⋯
L(s)  = 1  − 3/2·4-s − 4.02·5-s + 3.40·7-s + 0.353·8-s − 2.71·11-s + 0.832·13-s + 3/4·16-s − 3.63·17-s + 6.03·20-s − 1.25·23-s + 33/5·25-s − 5.10·28-s + 2.78·29-s − 13.6·35-s − 0.986·37-s − 1.42·40-s − 0.937·41-s + 2.28·43-s + 4.07·44-s − 1.31·47-s + 24/7·49-s − 1.24·52-s + 0.824·53-s + 10.9·55-s + 1.20·56-s + 1.95·59-s + 0.384·61-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(3.04902\times 10^{8}\)
Root analytic conductor: \(5.09346\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.589663251\)
\(L(\frac12)\) \(\approx\) \(1.589663251\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + 3 T^{2} - T^{3} + 3 p T^{4} - 3 p T^{5} + p^{3} T^{6} - 3 p^{2} T^{7} + 3 p^{3} T^{8} - p^{3} T^{9} + 3 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + 9 T + 48 T^{2} + 188 T^{3} + 609 T^{4} + 339 p T^{5} + 4076 T^{6} + 339 p^{2} T^{7} + 609 p^{2} T^{8} + 188 p^{3} T^{9} + 48 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 9 T + 57 T^{2} - 36 p T^{3} + 984 T^{4} - 3186 T^{5} + 9257 T^{6} - 3186 p T^{7} + 984 p^{2} T^{8} - 36 p^{4} T^{9} + 57 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 9 T + 72 T^{2} + 324 T^{3} + 1413 T^{4} + 4311 T^{5} + 15868 T^{6} + 4311 p T^{7} + 1413 p^{2} T^{8} + 324 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 66 T^{2} - 12 p T^{3} + 1929 T^{4} - 3657 T^{5} + 32297 T^{6} - 3657 p T^{7} + 1929 p^{2} T^{8} - 12 p^{4} T^{9} + 66 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 15 T + 138 T^{2} + 900 T^{3} + 4983 T^{4} + 24117 T^{5} + 105892 T^{6} + 24117 p T^{7} + 4983 p^{2} T^{8} + 900 p^{3} T^{9} + 138 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 99 T^{2} + 395 T^{3} + 4065 T^{4} + 12417 T^{5} + 108158 T^{6} + 12417 p T^{7} + 4065 p^{2} T^{8} + 395 p^{3} T^{9} + 99 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 15 T + 249 T^{2} - 2300 T^{3} + 21231 T^{4} - 136605 T^{5} + 861062 T^{6} - 136605 p T^{7} + 21231 p^{2} T^{8} - 2300 p^{3} T^{9} + 249 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 150 T^{2} - 10 T^{3} + 10266 T^{4} - 750 T^{5} + 407343 T^{6} - 750 p T^{7} + 10266 p^{2} T^{8} - 10 p^{3} T^{9} + 150 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 6 T + 138 T^{2} + 248 T^{3} + 5688 T^{4} - 14688 T^{5} + 146841 T^{6} - 14688 p T^{7} + 5688 p^{2} T^{8} + 248 p^{3} T^{9} + 138 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 159 T^{2} + 635 T^{3} + 11889 T^{4} + 35895 T^{5} + 577886 T^{6} + 35895 p T^{7} + 11889 p^{2} T^{8} + 635 p^{3} T^{9} + 159 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 15 T + 192 T^{2} - 1678 T^{3} + 15297 T^{4} - 108039 T^{5} + 788787 T^{6} - 108039 p T^{7} + 15297 p^{2} T^{8} - 1678 p^{3} T^{9} + 192 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T + 6 p T^{2} + 1926 T^{3} + 32541 T^{4} + 171513 T^{5} + 2020408 T^{6} + 171513 p T^{7} + 32541 p^{2} T^{8} + 1926 p^{3} T^{9} + 6 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T + 219 T^{2} - 1323 T^{3} + 21777 T^{4} - 126951 T^{5} + 1374406 T^{6} - 126951 p T^{7} + 21777 p^{2} T^{8} - 1323 p^{3} T^{9} + 219 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 15 T + 378 T^{2} - 3870 T^{3} + 55179 T^{4} - 421431 T^{5} + 4291516 T^{6} - 421431 p T^{7} + 55179 p^{2} T^{8} - 3870 p^{3} T^{9} + 378 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T + 213 T^{2} - 584 T^{3} + 21186 T^{4} - 52218 T^{5} + 1453581 T^{6} - 52218 p T^{7} + 21186 p^{2} T^{8} - 584 p^{3} T^{9} + 213 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 18 T + 357 T^{2} + 4373 T^{3} + 54933 T^{4} + 510171 T^{5} + 4792818 T^{6} + 510171 p T^{7} + 54933 p^{2} T^{8} + 4373 p^{3} T^{9} + 357 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 15 T + 387 T^{2} - 4446 T^{3} + 64233 T^{4} - 581379 T^{5} + 5950702 T^{6} - 581379 p T^{7} + 64233 p^{2} T^{8} - 4446 p^{3} T^{9} + 387 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T + 300 T^{2} + 2038 T^{3} + 43782 T^{4} + 286698 T^{5} + 3940299 T^{6} + 286698 p T^{7} + 43782 p^{2} T^{8} + 2038 p^{3} T^{9} + 300 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 21 T + 420 T^{2} - 6402 T^{3} + 81501 T^{4} - 888861 T^{5} + 8552927 T^{6} - 888861 p T^{7} + 81501 p^{2} T^{8} - 6402 p^{3} T^{9} + 420 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 3 T + 222 T^{2} - 1246 T^{3} + 34209 T^{4} - 157407 T^{5} + 3488216 T^{6} - 157407 p T^{7} + 34209 p^{2} T^{8} - 1246 p^{3} T^{9} + 222 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 24 T + 531 T^{2} - 7099 T^{3} + 93819 T^{4} - 932253 T^{5} + 9844202 T^{6} - 932253 p T^{7} + 93819 p^{2} T^{8} - 7099 p^{3} T^{9} + 531 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T + 306 T^{2} - 110 T^{3} + 38589 T^{4} - 171843 T^{5} + 3552945 T^{6} - 171843 p T^{7} + 38589 p^{2} T^{8} - 110 p^{3} T^{9} + 306 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.52837632532537914561926032364, −4.18098852593814417692949916185, −4.16950369723503647391096219840, −4.15174944428384548647191405221, −4.04957377692539991775802602563, −3.89988384817781274614588125400, −3.88289822940206521358046546708, −3.46407479826370663858143710620, −3.34090327637868914398046435503, −3.15500086159515637461488108507, −3.12393696986281447811087766738, −2.85074092958504843625581665488, −2.62614375693512571864523958541, −2.49792769274874644102540246297, −2.26078311547228805571877520285, −2.15620988222832895289773884672, −1.81317685750476344267552846072, −1.79695876910324044826080663272, −1.65152807983114834177456861034, −1.59161631580643710049613733889, −0.77121093912341004393197042947, −0.73908812342758806122456910870, −0.55143207021196754254340473456, −0.45405320317651758866601700947, −0.32146909192120586736141018689, 0.32146909192120586736141018689, 0.45405320317651758866601700947, 0.55143207021196754254340473456, 0.73908812342758806122456910870, 0.77121093912341004393197042947, 1.59161631580643710049613733889, 1.65152807983114834177456861034, 1.79695876910324044826080663272, 1.81317685750476344267552846072, 2.15620988222832895289773884672, 2.26078311547228805571877520285, 2.49792769274874644102540246297, 2.62614375693512571864523958541, 2.85074092958504843625581665488, 3.12393696986281447811087766738, 3.15500086159515637461488108507, 3.34090327637868914398046435503, 3.46407479826370663858143710620, 3.88289822940206521358046546708, 3.89988384817781274614588125400, 4.04957377692539991775802602563, 4.15174944428384548647191405221, 4.16950369723503647391096219840, 4.18098852593814417692949916185, 4.52837632532537914561926032364

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

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