Properties

Label 2-162-1.1-c5-0-12
Degree $2$
Conductor $162$
Sign $-1$
Analytic cond. $25.9821$
Root an. cond. $5.09727$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

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

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(25.9821\)
Root analytic conductor: \(5.09727\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 162,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line