Properties

Label 2-2151-1.1-c1-0-74
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  + 0.899·2-s − 1.19·4-s − 1.67·5-s − 1.75·7-s − 2.87·8-s − 1.50·10-s + 5.08·11-s + 6.43·13-s − 1.58·14-s − 0.201·16-s + 2.24·17-s − 3.96·19-s + 1.99·20-s + 4.57·22-s − 7.71·23-s − 2.19·25-s + 5.79·26-s + 2.09·28-s − 0.101·29-s − 4.63·31-s + 5.55·32-s + 2.01·34-s + 2.94·35-s − 7.78·37-s − 3.56·38-s + 4.80·40-s − 1.42·41-s + ⋯
L(s)  = 1  + 0.636·2-s − 0.595·4-s − 0.748·5-s − 0.664·7-s − 1.01·8-s − 0.476·10-s + 1.53·11-s + 1.78·13-s − 0.422·14-s − 0.0504·16-s + 0.543·17-s − 0.908·19-s + 0.445·20-s + 0.976·22-s − 1.60·23-s − 0.439·25-s + 1.13·26-s + 0.395·28-s − 0.0188·29-s − 0.832·31-s + 0.982·32-s + 0.345·34-s + 0.497·35-s − 1.27·37-s − 0.578·38-s + 0.759·40-s − 0.222·41-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 \( 1 - 0.899T + 2T^{2} \)
5 \( 1 + 1.67T + 5T^{2} \)
7 \( 1 + 1.75T + 7T^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 - 6.43T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 3.96T + 19T^{2} \)
23 \( 1 + 7.71T + 23T^{2} \)
29 \( 1 + 0.101T + 29T^{2} \)
31 \( 1 + 4.63T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + 1.42T + 41T^{2} \)
43 \( 1 + 3.65T + 43T^{2} \)
47 \( 1 - 7.73T + 47T^{2} \)
53 \( 1 + 1.61T + 53T^{2} \)
59 \( 1 - 2.46T + 59T^{2} \)
61 \( 1 + 4.77T + 61T^{2} \)
67 \( 1 + 3.42T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 - 0.423T + 83T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 - 6.06T + 97T^{2} \)
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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

−8.652509837680412428708847280399, −8.145961070990418909055293572800, −6.90074473282054767965109109121, −6.16849316257898540005045346355, −5.65526709536943049566243859939, −4.19500101962515673629936425440, −3.90097865914353830232270829547, −3.30760451202736388203106044519, −1.52783842239567128542582705179, 0, 1.52783842239567128542582705179, 3.30760451202736388203106044519, 3.90097865914353830232270829547, 4.19500101962515673629936425440, 5.65526709536943049566243859939, 6.16849316257898540005045346355, 6.90074473282054767965109109121, 8.145961070990418909055293572800, 8.652509837680412428708847280399

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