Properties

Label 2-2151-1.1-c1-0-67
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0242·2-s − 1.99·4-s + 2.24·5-s − 4.21·7-s + 0.0969·8-s − 0.0544·10-s + 2.12·11-s − 3.04·13-s + 0.102·14-s + 3.99·16-s + 4.57·17-s − 0.932·19-s − 4.48·20-s − 0.0515·22-s + 4.32·23-s + 0.0373·25-s + 0.0739·26-s + 8.42·28-s + 5.55·29-s − 1.27·31-s − 0.290·32-s − 0.110·34-s − 9.46·35-s − 8.37·37-s + 0.0226·38-s + 0.217·40-s − 10.8·41-s + ⋯
L(s)  = 1  − 0.0171·2-s − 0.999·4-s + 1.00·5-s − 1.59·7-s + 0.0342·8-s − 0.0172·10-s + 0.640·11-s − 0.845·13-s + 0.0273·14-s + 0.999·16-s + 1.10·17-s − 0.213·19-s − 1.00·20-s − 0.0109·22-s + 0.902·23-s + 0.00747·25-s + 0.0144·26-s + 1.59·28-s + 1.03·29-s − 0.228·31-s − 0.0514·32-s − 0.0190·34-s − 1.59·35-s − 1.37·37-s + 0.00366·38-s + 0.0344·40-s − 1.69·41-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: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.0242T + 2T^{2} \)
5 \( 1 - 2.24T + 5T^{2} \)
7 \( 1 + 4.21T + 7T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 + 3.04T + 13T^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 + 0.932T + 19T^{2} \)
23 \( 1 - 4.32T + 23T^{2} \)
29 \( 1 - 5.55T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 + 8.37T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 9.30T + 43T^{2} \)
47 \( 1 - 7.05T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 4.93T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 5.92T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 4.19T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 3.59T + 97T^{2} \)
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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

−9.114929991666507758409761411460, −8.033652772777948517371381843972, −6.93038574457889648864161901258, −6.34091235094143457305772225221, −5.48361506391999639877438303807, −4.79214004704421302960095446283, −3.56099130262708696090516007884, −2.97965691777691923099123598416, −1.47315085461701331852602366454, 0, 1.47315085461701331852602366454, 2.97965691777691923099123598416, 3.56099130262708696090516007884, 4.79214004704421302960095446283, 5.48361506391999639877438303807, 6.34091235094143457305772225221, 6.93038574457889648864161901258, 8.033652772777948517371381843972, 9.114929991666507758409761411460

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