Properties

Label 2-2151-1.1-c1-0-30
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  − 2.39·2-s + 3.74·4-s − 4.10·5-s − 3.62·7-s − 4.16·8-s + 9.83·10-s − 1.86·11-s + 3.35·13-s + 8.68·14-s + 2.50·16-s − 0.173·17-s − 1.09·19-s − 15.3·20-s + 4.47·22-s + 5.06·23-s + 11.8·25-s − 8.04·26-s − 13.5·28-s − 6.94·29-s + 6.11·31-s + 2.32·32-s + 0.414·34-s + 14.8·35-s + 3.44·37-s + 2.63·38-s + 17.1·40-s + 5.66·41-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.87·4-s − 1.83·5-s − 1.36·7-s − 1.47·8-s + 3.11·10-s − 0.562·11-s + 0.931·13-s + 2.32·14-s + 0.627·16-s − 0.0419·17-s − 0.251·19-s − 3.43·20-s + 0.953·22-s + 1.05·23-s + 2.37·25-s − 1.57·26-s − 2.56·28-s − 1.28·29-s + 1.09·31-s + 0.411·32-s + 0.0711·34-s + 2.51·35-s + 0.566·37-s + 0.426·38-s + 2.70·40-s + 0.884·41-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 \( 1 + 2.39T + 2T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
7 \( 1 + 3.62T + 7T^{2} \)
11 \( 1 + 1.86T + 11T^{2} \)
13 \( 1 - 3.35T + 13T^{2} \)
17 \( 1 + 0.173T + 17T^{2} \)
19 \( 1 + 1.09T + 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 + 6.94T + 29T^{2} \)
31 \( 1 - 6.11T + 31T^{2} \)
37 \( 1 - 3.44T + 37T^{2} \)
41 \( 1 - 5.66T + 41T^{2} \)
43 \( 1 + 5.01T + 43T^{2} \)
47 \( 1 - 5.07T + 47T^{2} \)
53 \( 1 - 4.55T + 53T^{2} \)
59 \( 1 - 4.23T + 59T^{2} \)
61 \( 1 - 8.14T + 61T^{2} \)
67 \( 1 + 8.96T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 6.34T + 73T^{2} \)
79 \( 1 - 1.38T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 17.5T + 97T^{2} \)
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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

−8.666358076307207797046322967114, −8.075739007463778443355623239318, −7.30492576795596168580588385733, −6.86518376043493362262885818217, −5.90796631945264906938652891154, −4.39545957353013051205485160300, −3.46552989095977371142076031870, −2.69869398543015423478471592652, −0.929756556361103899581368626585, 0, 0.929756556361103899581368626585, 2.69869398543015423478471592652, 3.46552989095977371142076031870, 4.39545957353013051205485160300, 5.90796631945264906938652891154, 6.86518376043493362262885818217, 7.30492576795596168580588385733, 8.075739007463778443355623239318, 8.666358076307207797046322967114

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