Properties

Label 2-2151-1.1-c1-0-57
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  − 1.61·2-s + 0.614·4-s − 1.46·5-s + 3.83·7-s + 2.24·8-s + 2.37·10-s − 2.90·11-s + 2.33·13-s − 6.20·14-s − 4.85·16-s − 3.77·17-s − 1.21·19-s − 0.901·20-s + 4.69·22-s + 1.47·23-s − 2.85·25-s − 3.76·26-s + 2.35·28-s − 1.02·29-s − 1.72·31-s + 3.36·32-s + 6.09·34-s − 5.62·35-s − 1.08·37-s + 1.96·38-s − 3.28·40-s + 1.58·41-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.307·4-s − 0.655·5-s + 1.44·7-s + 0.791·8-s + 0.749·10-s − 0.875·11-s + 0.646·13-s − 1.65·14-s − 1.21·16-s − 0.914·17-s − 0.279·19-s − 0.201·20-s + 1.00·22-s + 0.307·23-s − 0.570·25-s − 0.739·26-s + 0.445·28-s − 0.190·29-s − 0.308·31-s + 0.594·32-s + 1.04·34-s − 0.950·35-s − 0.179·37-s + 0.319·38-s − 0.519·40-s + 0.246·41-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 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 + 1.46T + 5T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 - 2.33T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
19 \( 1 + 1.21T + 19T^{2} \)
23 \( 1 - 1.47T + 23T^{2} \)
29 \( 1 + 1.02T + 29T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 + 1.08T + 37T^{2} \)
41 \( 1 - 1.58T + 41T^{2} \)
43 \( 1 - 0.433T + 43T^{2} \)
47 \( 1 + 7.65T + 47T^{2} \)
53 \( 1 + 1.07T + 53T^{2} \)
59 \( 1 - 4.82T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 6.11T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 6.08T + 89T^{2} \)
97 \( 1 - 7.99T + 97T^{2} \)
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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

−8.587102340578765836069106283239, −8.000242154919240874909105215855, −7.61549174006609877732476472577, −6.66129794722342781777067522628, −5.37179270836595936558916040011, −4.64360366605365318322439112610, −3.87061547662322498004781917566, −2.32645096457765236043841735274, −1.37440404473971757334133741069, 0, 1.37440404473971757334133741069, 2.32645096457765236043841735274, 3.87061547662322498004781917566, 4.64360366605365318322439112610, 5.37179270836595936558916040011, 6.66129794722342781777067522628, 7.61549174006609877732476472577, 8.000242154919240874909105215855, 8.587102340578765836069106283239

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