Properties

Label 2-2151-1.1-c1-0-59
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  + 0.510·2-s − 1.73·4-s − 1.97·5-s − 1.30·7-s − 1.90·8-s − 1.00·10-s + 4.91·11-s + 1.24·13-s − 0.667·14-s + 2.50·16-s + 0.353·17-s + 3.33·19-s + 3.43·20-s + 2.51·22-s + 2.97·23-s − 1.10·25-s + 0.635·26-s + 2.27·28-s − 7.75·29-s − 3.52·31-s + 5.09·32-s + 0.180·34-s + 2.58·35-s + 7.67·37-s + 1.70·38-s + 3.76·40-s − 10.3·41-s + ⋯
L(s)  = 1  + 0.360·2-s − 0.869·4-s − 0.882·5-s − 0.494·7-s − 0.674·8-s − 0.318·10-s + 1.48·11-s + 0.345·13-s − 0.178·14-s + 0.626·16-s + 0.0857·17-s + 0.765·19-s + 0.767·20-s + 0.535·22-s + 0.620·23-s − 0.221·25-s + 0.124·26-s + 0.430·28-s − 1.44·29-s − 0.633·31-s + 0.900·32-s + 0.0309·34-s + 0.436·35-s + 1.26·37-s + 0.276·38-s + 0.595·40-s − 1.61·41-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 \( 1 - 0.510T + 2T^{2} \)
5 \( 1 + 1.97T + 5T^{2} \)
7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
17 \( 1 - 0.353T + 17T^{2} \)
19 \( 1 - 3.33T + 19T^{2} \)
23 \( 1 - 2.97T + 23T^{2} \)
29 \( 1 + 7.75T + 29T^{2} \)
31 \( 1 + 3.52T + 31T^{2} \)
37 \( 1 - 7.67T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 3.54T + 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 - 1.32T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 4.16T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 3.43T + 89T^{2} \)
97 \( 1 - 0.321T + 97T^{2} \)
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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

−8.734217332809192974593285996525, −8.010247480153398646730331158143, −7.09007724829435392446301332004, −6.30932156425159223454981334553, −5.40542872516224437018033438093, −4.50399105614027810704496415950, −3.63468283918103844565047752896, −3.33989362388299408072845134230, −1.40727071680839315537375208160, 0, 1.40727071680839315537375208160, 3.33989362388299408072845134230, 3.63468283918103844565047752896, 4.50399105614027810704496415950, 5.40542872516224437018033438093, 6.30932156425159223454981334553, 7.09007724829435392446301332004, 8.010247480153398646730331158143, 8.734217332809192974593285996525

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