Properties

Label 2-2151-1.1-c1-0-82
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  + 0.921·2-s − 1.15·4-s − 1.07·5-s + 3.99·7-s − 2.90·8-s − 0.991·10-s − 3.85·11-s + 1.10·13-s + 3.68·14-s − 0.370·16-s − 4.98·17-s + 6.77·19-s + 1.24·20-s − 3.55·22-s − 4.48·23-s − 3.84·25-s + 1.01·26-s − 4.60·28-s + 0.455·29-s − 0.141·31-s + 5.46·32-s − 4.58·34-s − 4.30·35-s − 5.62·37-s + 6.23·38-s + 3.12·40-s − 3.87·41-s + ⋯
L(s)  = 1  + 0.651·2-s − 0.575·4-s − 0.481·5-s + 1.51·7-s − 1.02·8-s − 0.313·10-s − 1.16·11-s + 0.306·13-s + 0.984·14-s − 0.0925·16-s − 1.20·17-s + 1.55·19-s + 0.277·20-s − 0.757·22-s − 0.935·23-s − 0.768·25-s + 0.199·26-s − 0.870·28-s + 0.0846·29-s − 0.0254·31-s + 0.966·32-s − 0.786·34-s − 0.727·35-s − 0.925·37-s + 1.01·38-s + 0.494·40-s − 0.605·41-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 \( 1 - 0.921T + 2T^{2} \)
5 \( 1 + 1.07T + 5T^{2} \)
7 \( 1 - 3.99T + 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 - 1.10T + 13T^{2} \)
17 \( 1 + 4.98T + 17T^{2} \)
19 \( 1 - 6.77T + 19T^{2} \)
23 \( 1 + 4.48T + 23T^{2} \)
29 \( 1 - 0.455T + 29T^{2} \)
31 \( 1 + 0.141T + 31T^{2} \)
37 \( 1 + 5.62T + 37T^{2} \)
41 \( 1 + 3.87T + 41T^{2} \)
43 \( 1 + 5.21T + 43T^{2} \)
47 \( 1 - 0.0217T + 47T^{2} \)
53 \( 1 - 8.05T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 8.40T + 67T^{2} \)
71 \( 1 + 1.78T + 71T^{2} \)
73 \( 1 - 8.35T + 73T^{2} \)
79 \( 1 + 16.7T + 79T^{2} \)
83 \( 1 - 3.41T + 83T^{2} \)
89 \( 1 - 7.49T + 89T^{2} \)
97 \( 1 + 9.46T + 97T^{2} \)
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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

−8.511623714018949411732014570405, −7.983352797847098788877848629463, −7.33004512116084406241202762018, −6.02977912775314505547746515309, −5.23421106485419568689792399012, −4.72665230419124240934630771206, −3.94993613842333357940492918179, −2.94075709323790718664743018462, −1.68871384069805730792205188197, 0, 1.68871384069805730792205188197, 2.94075709323790718664743018462, 3.94993613842333357940492918179, 4.72665230419124240934630771206, 5.23421106485419568689792399012, 6.02977912775314505547746515309, 7.33004512116084406241202762018, 7.983352797847098788877848629463, 8.511623714018949411732014570405

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