Properties

Label 2-2151-1.1-c1-0-87
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  + 1.25·2-s − 0.430·4-s + 1.67·5-s − 0.437·7-s − 3.04·8-s + 2.10·10-s − 3.01·11-s + 3.79·13-s − 0.548·14-s − 2.95·16-s − 4.87·17-s − 7.63·19-s − 0.721·20-s − 3.78·22-s − 0.261·23-s − 2.18·25-s + 4.75·26-s + 0.188·28-s − 0.689·29-s + 8.79·31-s + 2.38·32-s − 6.10·34-s − 0.734·35-s − 6.63·37-s − 9.56·38-s − 5.10·40-s + 0.687·41-s + ⋯
L(s)  = 1  + 0.885·2-s − 0.215·4-s + 0.750·5-s − 0.165·7-s − 1.07·8-s + 0.664·10-s − 0.909·11-s + 1.05·13-s − 0.146·14-s − 0.738·16-s − 1.18·17-s − 1.75·19-s − 0.161·20-s − 0.806·22-s − 0.0544·23-s − 0.436·25-s + 0.933·26-s + 0.0355·28-s − 0.128·29-s + 1.58·31-s + 0.422·32-s − 1.04·34-s − 0.124·35-s − 1.09·37-s − 1.55·38-s − 0.807·40-s + 0.107·41-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 \( 1 - 1.25T + 2T^{2} \)
5 \( 1 - 1.67T + 5T^{2} \)
7 \( 1 + 0.437T + 7T^{2} \)
11 \( 1 + 3.01T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 + 4.87T + 17T^{2} \)
19 \( 1 + 7.63T + 19T^{2} \)
23 \( 1 + 0.261T + 23T^{2} \)
29 \( 1 + 0.689T + 29T^{2} \)
31 \( 1 - 8.79T + 31T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 - 0.687T + 41T^{2} \)
43 \( 1 + 7.99T + 43T^{2} \)
47 \( 1 + 0.785T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 1.44T + 61T^{2} \)
67 \( 1 - 6.28T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 9.10T + 73T^{2} \)
79 \( 1 - 5.17T + 79T^{2} \)
83 \( 1 - 1.59T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 - 2.60T + 97T^{2} \)
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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

−8.554218522976915303148068007511, −8.223311354694878158105687710895, −6.56711321328890264265796970563, −6.36118885799646137440470090571, −5.42088457532519989770710609922, −4.66459474508976454421927998315, −3.89028264673573835317876141096, −2.85362075391586919287242605287, −1.91475072444509226377021114986, 0, 1.91475072444509226377021114986, 2.85362075391586919287242605287, 3.89028264673573835317876141096, 4.66459474508976454421927998315, 5.42088457532519989770710609922, 6.36118885799646137440470090571, 6.56711321328890264265796970563, 8.223311354694878158105687710895, 8.554218522976915303148068007511

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