Properties

Label 2-1151-1151.1150-c0-0-15
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  + 0.380·2-s + 1.90·3-s − 0.855·4-s + 0.0766·5-s + 0.726·6-s − 1.33·7-s − 0.706·8-s + 2.63·9-s + 0.0291·10-s + 1.79·11-s − 1.63·12-s − 0.506·14-s + 0.146·15-s + 0.586·16-s + 1.00·18-s − 0.0654·20-s − 2.53·21-s + 0.682·22-s − 1.34·24-s − 0.994·25-s + 3.11·27-s + 1.13·28-s − 1.94·29-s + 0.0556·30-s + 0.929·32-s + 3.41·33-s − 0.101·35-s + ⋯
L(s)  = 1  + 0.380·2-s + 1.90·3-s − 0.855·4-s + 0.0766·5-s + 0.726·6-s − 1.33·7-s − 0.706·8-s + 2.63·9-s + 0.0291·10-s + 1.79·11-s − 1.63·12-s − 0.506·14-s + 0.146·15-s + 0.586·16-s + 1.00·18-s − 0.0654·20-s − 2.53·21-s + 0.682·22-s − 1.34·24-s − 0.994·25-s + 3.11·27-s + 1.13·28-s − 1.94·29-s + 0.0556·30-s + 0.929·32-s + 3.41·33-s − 0.101·35-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

\(L(\frac{1}{2})\) \(\approx\) \(1.822089312\)
\(L(\frac12)\) \(\approx\) \(1.822089312\)
\(L(1)\) 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 \( 1 - 0.380T + T^{2} \)
3 \( 1 - 1.90T + T^{2} \)
5 \( 1 - 0.0766T + T^{2} \)
7 \( 1 + 1.33T + T^{2} \)
11 \( 1 - 1.79T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.94T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.54T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.08T + T^{2} \)
47 \( 1 - 1.97T + T^{2} \)
53 \( 1 - 0.955T + T^{2} \)
59 \( 1 + 1.99T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.71T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.229T + T^{2} \)
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

−9.564061622103492956819259851540, −9.139920246060047542083903163390, −8.758515540627315180044721805470, −7.59938202790995349419882214499, −6.81304551709284438009563972955, −5.82796075586274958603373734139, −4.26834767264519063111684318160, −3.70743779098888958826730707371, −3.17362214526586055163500483110, −1.74418135004610338397050114911, 1.74418135004610338397050114911, 3.17362214526586055163500483110, 3.70743779098888958826730707371, 4.26834767264519063111684318160, 5.82796075586274958603373734139, 6.81304551709284438009563972955, 7.59938202790995349419882214499, 8.758515540627315180044721805470, 9.139920246060047542083903163390, 9.564061622103492956819259851540

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