Properties

Label 2-1151-1151.1150-c0-0-13
Degree $2$
Conductor $1151$
Sign $1$
Analytic cond. $0.574423$
Root an. cond. $0.757907$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54·2-s + 1.97·3-s + 1.38·4-s + 1.44·5-s − 3.04·6-s − 0.818·7-s − 0.587·8-s + 2.90·9-s − 2.22·10-s − 1.94·11-s + 2.72·12-s + 1.26·14-s + 2.84·15-s − 0.474·16-s − 4.48·18-s + 1.98·20-s − 1.61·21-s + 3.00·22-s − 1.16·24-s + 1.07·25-s + 3.76·27-s − 1.12·28-s − 0.229·29-s − 4.39·30-s + 1.31·32-s − 3.84·33-s − 1.17·35-s + ⋯
L(s)  = 1  − 1.54·2-s + 1.97·3-s + 1.38·4-s + 1.44·5-s − 3.04·6-s − 0.818·7-s − 0.587·8-s + 2.90·9-s − 2.22·10-s − 1.94·11-s + 2.72·12-s + 1.26·14-s + 2.84·15-s − 0.474·16-s − 4.48·18-s + 1.98·20-s − 1.61·21-s + 3.00·22-s − 1.16·24-s + 1.07·25-s + 3.76·27-s − 1.12·28-s − 0.229·29-s − 4.39·30-s + 1.31·32-s − 3.84·33-s − 1.17·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1151\)
Sign: $1$
Analytic conductor: \(0.574423\)
Root analytic conductor: \(0.757907\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1151} (1150, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1151,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078505679\)
\(L(\frac12)\) \(\approx\) \(1.078505679\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1151 \( 1 - T \)
good2 \( 1 + 1.54T + T^{2} \)
3 \( 1 - 1.97T + T^{2} \)
5 \( 1 - 1.44T + T^{2} \)
7 \( 1 + 0.818T + T^{2} \)
11 \( 1 + 1.94T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 0.229T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.676T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.955T + T^{2} \)
47 \( 1 + 1.99T + T^{2} \)
53 \( 1 + 1.71T + T^{2} \)
59 \( 1 - 0.0766T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.529T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.33T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.747326391434078590997297766536, −9.334206097216429184821632174820, −8.464325720629819381405963240934, −7.88446790609264200966899479631, −7.17789708275969238427277351754, −6.19796338862628704395842553067, −4.77974097110852192866667936544, −3.10102448780345108918181793998, −2.48599596464021658242184605788, −1.67491823708135153088302467569, 1.67491823708135153088302467569, 2.48599596464021658242184605788, 3.10102448780345108918181793998, 4.77974097110852192866667936544, 6.19796338862628704395842553067, 7.17789708275969238427277351754, 7.88446790609264200966899479631, 8.464325720629819381405963240934, 9.334206097216429184821632174820, 9.747326391434078590997297766536

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