Properties

Label 2-2151-1.1-c1-0-3
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0842·2-s − 1.99·4-s − 2.54·5-s − 1.62·7-s − 0.336·8-s − 0.214·10-s − 5.58·11-s + 1.97·13-s − 0.136·14-s + 3.95·16-s − 6.03·17-s − 3.46·19-s + 5.06·20-s − 0.470·22-s + 0.922·23-s + 1.46·25-s + 0.166·26-s + 3.23·28-s − 6.23·29-s + 0.733·31-s + 1.00·32-s − 0.508·34-s + 4.13·35-s − 5.69·37-s − 0.292·38-s + 0.855·40-s − 7.79·41-s + ⋯
L(s)  = 1  + 0.0595·2-s − 0.996·4-s − 1.13·5-s − 0.614·7-s − 0.118·8-s − 0.0677·10-s − 1.68·11-s + 0.549·13-s − 0.0365·14-s + 0.989·16-s − 1.46·17-s − 0.795·19-s + 1.13·20-s − 0.100·22-s + 0.192·23-s + 0.292·25-s + 0.0327·26-s + 0.612·28-s − 1.15·29-s + 0.131·31-s + 0.177·32-s − 0.0872·34-s + 0.698·35-s − 0.936·37-s − 0.0473·38-s + 0.135·40-s − 1.21·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3132131853\)
\(L(\frac12)\) \(\approx\) \(0.3132131853\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 - 0.0842T + 2T^{2} \)
5 \( 1 + 2.54T + 5T^{2} \)
7 \( 1 + 1.62T + 7T^{2} \)
11 \( 1 + 5.58T + 11T^{2} \)
13 \( 1 - 1.97T + 13T^{2} \)
17 \( 1 + 6.03T + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 - 0.922T + 23T^{2} \)
29 \( 1 + 6.23T + 29T^{2} \)
31 \( 1 - 0.733T + 31T^{2} \)
37 \( 1 + 5.69T + 37T^{2} \)
41 \( 1 + 7.79T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 - 9.19T + 53T^{2} \)
59 \( 1 + 9.53T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 - 8.45T + 67T^{2} \)
71 \( 1 + 5.78T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + 0.759T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 8.41T + 89T^{2} \)
97 \( 1 + 6.97T + 97T^{2} \)
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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

−8.859994667658038441406269425890, −8.407994301241513499840226350945, −7.66998027767061490657045582218, −6.86026075952917373011424038249, −5.77365070291670195193163713265, −4.97351305382888260355363830369, −4.11513408432067462820331954306, −3.51998241456158742308772272664, −2.35926183220384618347775097837, −0.34386423185050138387792423129, 0.34386423185050138387792423129, 2.35926183220384618347775097837, 3.51998241456158742308772272664, 4.11513408432067462820331954306, 4.97351305382888260355363830369, 5.77365070291670195193163713265, 6.86026075952917373011424038249, 7.66998027767061490657045582218, 8.407994301241513499840226350945, 8.859994667658038441406269425890

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