Properties

Label 2-2151-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.12·2-s + 2.52·4-s + 2.10·5-s + 0.789·7-s − 1.12·8-s − 4.47·10-s + 4.48·11-s − 4.31·13-s − 1.67·14-s − 2.66·16-s − 6.64·17-s + 3.39·19-s + 5.31·20-s − 9.55·22-s + 9.17·23-s − 0.581·25-s + 9.17·26-s + 1.99·28-s + 0.384·29-s + 2.78·31-s + 7.92·32-s + 14.1·34-s + 1.65·35-s − 0.131·37-s − 7.23·38-s − 2.35·40-s + 8.64·41-s + ⋯
L(s)  = 1  − 1.50·2-s + 1.26·4-s + 0.940·5-s + 0.298·7-s − 0.396·8-s − 1.41·10-s + 1.35·11-s − 1.19·13-s − 0.448·14-s − 0.667·16-s − 1.61·17-s + 0.779·19-s + 1.18·20-s − 2.03·22-s + 1.91·23-s − 0.116·25-s + 1.79·26-s + 0.377·28-s + 0.0713·29-s + 0.500·31-s + 1.40·32-s + 2.42·34-s + 0.280·35-s − 0.0216·37-s − 1.17·38-s − 0.372·40-s + 1.35·41-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: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.058826911\)
\(L(\frac12)\) \(\approx\) \(1.058826911\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.12T + 2T^{2} \)
5 \( 1 - 2.10T + 5T^{2} \)
7 \( 1 - 0.789T + 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + 4.31T + 13T^{2} \)
17 \( 1 + 6.64T + 17T^{2} \)
19 \( 1 - 3.39T + 19T^{2} \)
23 \( 1 - 9.17T + 23T^{2} \)
29 \( 1 - 0.384T + 29T^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 + 0.131T + 37T^{2} \)
41 \( 1 - 8.64T + 41T^{2} \)
43 \( 1 + 3.94T + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 - 0.625T + 53T^{2} \)
59 \( 1 + 7.17T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 - 0.301T + 67T^{2} \)
71 \( 1 - 2.81T + 71T^{2} \)
73 \( 1 + 2.28T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.294671928488477501445550642900, −8.628328721738879871885855696145, −7.55288803016333901579541492849, −6.94034662282710715423997879061, −6.32458678745061433789566563735, −5.12901006946666156471699249183, −4.31058553659972561675242286793, −2.70963088051182822992006957285, −1.88079864176721001076213495850, −0.880813547125327509298831050375, 0.880813547125327509298831050375, 1.88079864176721001076213495850, 2.70963088051182822992006957285, 4.31058553659972561675242286793, 5.12901006946666156471699249183, 6.32458678745061433789566563735, 6.94034662282710715423997879061, 7.55288803016333901579541492849, 8.628328721738879871885855696145, 9.294671928488477501445550642900

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