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Properties

Label 2-162-1.1-c5-0-12
Degree 22
Conductor 162162
Sign −1-1
Analytic cond. 25.982125.9821
Root an. cond. 5.097275.09727
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s − 21·5-s + 74·7-s − 64·8-s + 84·10-s − 270·11-s − 115·13-s − 296·14-s + 256·16-s + 861·17-s + 1.85e3·19-s − 336·20-s + 1.08e3·22-s − 3.61e3·23-s − 2.68e3·25-s + 460·26-s + 1.18e3·28-s − 1.12e3·29-s + 5.22e3·31-s − 1.02e3·32-s − 3.44e3·34-s − 1.55e3·35-s + 9.91e3·37-s − 7.40e3·38-s + 1.34e3·40-s − 1.07e4·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.375·5-s + 0.570·7-s − 0.353·8-s + 0.265·10-s − 0.672·11-s − 0.188·13-s − 0.403·14-s + 1/4·16-s + 0.722·17-s + 1.17·19-s − 0.187·20-s + 0.475·22-s − 1.42·23-s − 0.858·25-s + 0.133·26-s + 0.285·28-s − 0.248·29-s + 0.977·31-s − 0.176·32-s − 0.510·34-s − 0.214·35-s + 1.19·37-s − 0.831·38-s + 0.132·40-s − 0.999·41-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −1-1
Analytic conductor: 25.982125.9821
Root analytic conductor: 5.097275.09727
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 162, ( :5/2), −1)(2,\ 162,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+p2T 1 + p^{2} T
3 1 1
good5 1+21T+p5T2 1 + 21 T + p^{5} T^{2}
7 1−74T+p5T2 1 - 74 T + p^{5} T^{2}
11 1+270T+p5T2 1 + 270 T + p^{5} T^{2}
13 1+115T+p5T2 1 + 115 T + p^{5} T^{2}
17 1−861T+p5T2 1 - 861 T + p^{5} T^{2}
19 1−1850T+p5T2 1 - 1850 T + p^{5} T^{2}
23 1+3618T+p5T2 1 + 3618 T + p^{5} T^{2}
29 1+1125T+p5T2 1 + 1125 T + p^{5} T^{2}
31 1−5228T+p5T2 1 - 5228 T + p^{5} T^{2}
37 1−9917T+p5T2 1 - 9917 T + p^{5} T^{2}
41 1+10758T+p5T2 1 + 10758 T + p^{5} T^{2}
43 1+19714T+p5T2 1 + 19714 T + p^{5} T^{2}
47 1+9984T+p5T2 1 + 9984 T + p^{5} T^{2}
53 1+36726T+p5T2 1 + 36726 T + p^{5} T^{2}
59 1+26460T+p5T2 1 + 26460 T + p^{5} T^{2}
61 1+53779T+p5T2 1 + 53779 T + p^{5} T^{2}
67 1+12934T+p5T2 1 + 12934 T + p^{5} T^{2}
71 1−4254T+p5T2 1 - 4254 T + p^{5} T^{2}
73 1+17521T+p5T2 1 + 17521 T + p^{5} T^{2}
79 1+36946T+p5T2 1 + 36946 T + p^{5} T^{2}
83 1−76416T+p5T2 1 - 76416 T + p^{5} T^{2}
89 1−45357T+p5T2 1 - 45357 T + p^{5} T^{2}
97 1−127574T+p5T2 1 - 127574 T + p^{5} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.51909936680503427975812663331, −10.29316234340885445789885747474, −9.540663835836650686316321655441, −7.997615749117124459344682799354, −7.77050153653225993151343386985, −6.16990717848624589674125705556, −4.86177132375486527833704145984, −3.20922041708012070632887187616, −1.61928064157313403467831788335, 0, 1.61928064157313403467831788335, 3.20922041708012070632887187616, 4.86177132375486527833704145984, 6.16990717848624589674125705556, 7.77050153653225993151343386985, 7.997615749117124459344682799354, 9.540663835836650686316321655441, 10.29316234340885445789885747474, 11.51909936680503427975812663331

Graph of the ZZ-function along the critical line