Properties

Label 2-2312-1.1-c3-0-1
Degree $2$
Conductor $2312$
Sign $1$
Analytic cond. $136.412$
Root an. cond. $11.6795$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.05·3-s − 8.16·5-s − 11.0·7-s + 9.61·9-s + 26.1·11-s − 34.2·13-s + 49.4·15-s − 130.·19-s + 66.6·21-s − 74.8·23-s − 58.2·25-s + 105.·27-s + 56.8·29-s − 90.6·31-s − 158.·33-s + 89.9·35-s − 206.·37-s + 207.·39-s + 171.·41-s − 169.·43-s − 78.4·45-s + 154.·47-s − 221.·49-s − 281.·53-s − 213.·55-s + 787.·57-s − 448.·59-s + ⋯
L(s)  = 1  − 1.16·3-s − 0.730·5-s − 0.594·7-s + 0.355·9-s + 0.716·11-s − 0.729·13-s + 0.850·15-s − 1.57·19-s + 0.692·21-s − 0.678·23-s − 0.466·25-s + 0.749·27-s + 0.363·29-s − 0.525·31-s − 0.833·33-s + 0.434·35-s − 0.918·37-s + 0.849·39-s + 0.651·41-s − 0.601·43-s − 0.260·45-s + 0.479·47-s − 0.646·49-s − 0.728·53-s − 0.523·55-s + 1.83·57-s − 0.989·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(136.412\)
Root analytic conductor: \(11.6795\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0001145920850\)
\(L(\frac12)\) \(\approx\) \(0.0001145920850\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + 6.05T + 27T^{2} \)
5 \( 1 + 8.16T + 125T^{2} \)
7 \( 1 + 11.0T + 343T^{2} \)
11 \( 1 - 26.1T + 1.33e3T^{2} \)
13 \( 1 + 34.2T + 2.19e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 + 74.8T + 1.21e4T^{2} \)
29 \( 1 - 56.8T + 2.43e4T^{2} \)
31 \( 1 + 90.6T + 2.97e4T^{2} \)
37 \( 1 + 206.T + 5.06e4T^{2} \)
41 \( 1 - 171.T + 6.89e4T^{2} \)
43 \( 1 + 169.T + 7.95e4T^{2} \)
47 \( 1 - 154.T + 1.03e5T^{2} \)
53 \( 1 + 281.T + 1.48e5T^{2} \)
59 \( 1 + 448.T + 2.05e5T^{2} \)
61 \( 1 + 232.T + 2.26e5T^{2} \)
67 \( 1 + 910.T + 3.00e5T^{2} \)
71 \( 1 + 661.T + 3.57e5T^{2} \)
73 \( 1 + 122.T + 3.89e5T^{2} \)
79 \( 1 - 135.T + 4.93e5T^{2} \)
83 \( 1 + 183.T + 5.71e5T^{2} \)
89 \( 1 + 776.T + 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−8.645290409519547686013782715670, −7.79622938264079192698407924659, −6.89100541746581445605668508466, −6.32801561927970816320726184229, −5.64048491210804854848063511117, −4.58837342224915350175048523434, −4.01933241660223928728202896423, −2.90776616642005814796343646905, −1.59545157147313305856480454404, −0.00502832029547353218602033198, 0.00502832029547353218602033198, 1.59545157147313305856480454404, 2.90776616642005814796343646905, 4.01933241660223928728202896423, 4.58837342224915350175048523434, 5.64048491210804854848063511117, 6.32801561927970816320726184229, 6.89100541746581445605668508466, 7.79622938264079192698407924659, 8.645290409519547686013782715670

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