Properties

Label 2-2151-1.1-c1-0-49
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  − 1.22·2-s − 0.505·4-s − 0.403·5-s − 4.45·7-s + 3.06·8-s + 0.492·10-s + 2.17·11-s + 5.85·13-s + 5.44·14-s − 2.73·16-s − 7.98·17-s + 2.00·19-s + 0.203·20-s − 2.65·22-s − 0.679·23-s − 4.83·25-s − 7.15·26-s + 2.25·28-s + 5.44·29-s − 0.595·31-s − 2.78·32-s + 9.75·34-s + 1.79·35-s + 5.86·37-s − 2.45·38-s − 1.23·40-s − 0.396·41-s + ⋯
L(s)  = 1  − 0.864·2-s − 0.252·4-s − 0.180·5-s − 1.68·7-s + 1.08·8-s + 0.155·10-s + 0.655·11-s + 1.62·13-s + 1.45·14-s − 0.683·16-s − 1.93·17-s + 0.459·19-s + 0.0456·20-s − 0.566·22-s − 0.141·23-s − 0.967·25-s − 1.40·26-s + 0.426·28-s + 1.01·29-s − 0.106·31-s − 0.492·32-s + 1.67·34-s + 0.303·35-s + 0.963·37-s − 0.397·38-s − 0.195·40-s − 0.0619·41-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 \( 1 + 1.22T + 2T^{2} \)
5 \( 1 + 0.403T + 5T^{2} \)
7 \( 1 + 4.45T + 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 - 5.85T + 13T^{2} \)
17 \( 1 + 7.98T + 17T^{2} \)
19 \( 1 - 2.00T + 19T^{2} \)
23 \( 1 + 0.679T + 23T^{2} \)
29 \( 1 - 5.44T + 29T^{2} \)
31 \( 1 + 0.595T + 31T^{2} \)
37 \( 1 - 5.86T + 37T^{2} \)
41 \( 1 + 0.396T + 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 - 6.38T + 47T^{2} \)
53 \( 1 + 9.69T + 53T^{2} \)
59 \( 1 + 2.87T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 7.74T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 6.99T + 83T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + 8.60T + 97T^{2} \)
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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

−8.886452626347241242641667707119, −8.173740386018844072226680583368, −7.13381521475683704669050488654, −6.44635468390002678192903226362, −5.84566827077755475420891905124, −4.27597203218920322425605976827, −3.89591949686488188911594713252, −2.68384457657599639687084849184, −1.20989923673633197383546031297, 0, 1.20989923673633197383546031297, 2.68384457657599639687084849184, 3.89591949686488188911594713252, 4.27597203218920322425605976827, 5.84566827077755475420891905124, 6.44635468390002678192903226362, 7.13381521475683704669050488654, 8.173740386018844072226680583368, 8.886452626347241242641667707119

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