Properties

Label 2-1151-1151.1150-c0-0-16
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  + 0.955·2-s + 1.79·3-s − 0.0871·4-s − 1.33·5-s + 1.71·6-s + 1.90·7-s − 1.03·8-s + 2.21·9-s − 1.27·10-s − 1.54·11-s − 0.156·12-s + 1.82·14-s − 2.38·15-s − 0.905·16-s + 2.11·18-s + 0.115·20-s + 3.41·21-s − 1.47·22-s − 1.86·24-s + 0.770·25-s + 2.17·27-s − 0.166·28-s + 0.676·29-s − 2.27·30-s + 0.173·32-s − 2.76·33-s − 2.53·35-s + ⋯
L(s)  = 1  + 0.955·2-s + 1.79·3-s − 0.0871·4-s − 1.33·5-s + 1.71·6-s + 1.90·7-s − 1.03·8-s + 2.21·9-s − 1.27·10-s − 1.54·11-s − 0.156·12-s + 1.82·14-s − 2.38·15-s − 0.905·16-s + 2.11·18-s + 0.115·20-s + 3.41·21-s − 1.47·22-s − 1.86·24-s + 0.770·25-s + 2.17·27-s − 0.166·28-s + 0.676·29-s − 2.27·30-s + 0.173·32-s − 2.76·33-s − 2.53·35-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

\(L(\frac{1}{2})\) \(\approx\) \(2.291408021\)
\(L(\frac12)\) \(\approx\) \(2.291408021\)
\(L(1)\) 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 \( 1 - 0.955T + T^{2} \)
3 \( 1 - 1.79T + T^{2} \)
5 \( 1 + 1.33T + T^{2} \)
7 \( 1 - 1.90T + T^{2} \)
11 \( 1 + 1.54T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.676T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.71T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.99T + T^{2} \)
47 \( 1 + 1.94T + T^{2} \)
53 \( 1 - 0.0766T + T^{2} \)
59 \( 1 + 0.229T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.44T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.63T + T^{2} \)
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

−9.940337485704121189587457490920, −8.679540308875644723427712373864, −8.225264337175558609772766189003, −7.934828455138790380756632730084, −7.00822060879165248225929562154, −5.07219976802220001600443484546, −4.76683768055546751946363478660, −3.78608654141864889910124530260, −3.08545181628854815361095443223, −1.97597329520505202385499367031, 1.97597329520505202385499367031, 3.08545181628854815361095443223, 3.78608654141864889910124530260, 4.76683768055546751946363478660, 5.07219976802220001600443484546, 7.00822060879165248225929562154, 7.934828455138790380756632730084, 8.225264337175558609772766189003, 8.679540308875644723427712373864, 9.940337485704121189587457490920

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