Properties

Label 2-2312-1.1-c3-0-13
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  − 9.26·3-s + 16.4·5-s − 34.5·7-s + 58.7·9-s − 7.42·11-s − 42.0·13-s − 151.·15-s − 59.9·19-s + 319.·21-s − 49.4·23-s + 144.·25-s − 294.·27-s − 259.·29-s − 92.2·31-s + 68.7·33-s − 566.·35-s + 207.·37-s + 389.·39-s − 176.·41-s + 19.0·43-s + 964.·45-s + 80.1·47-s + 847.·49-s − 319.·53-s − 121.·55-s + 554.·57-s − 11.0·59-s + ⋯
L(s)  = 1  − 1.78·3-s + 1.46·5-s − 1.86·7-s + 2.17·9-s − 0.203·11-s − 0.896·13-s − 2.61·15-s − 0.723·19-s + 3.32·21-s − 0.448·23-s + 1.15·25-s − 2.09·27-s − 1.66·29-s − 0.534·31-s + 0.362·33-s − 2.73·35-s + 0.922·37-s + 1.59·39-s − 0.673·41-s + 0.0676·43-s + 3.19·45-s + 0.248·47-s + 2.47·49-s − 0.829·53-s − 0.298·55-s + 1.28·57-s − 0.0244·59-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: $\chi_{2312} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2979774806\)
\(L(\frac12)\) \(\approx\) \(0.2979774806\)
\(L(\frac{5}{2})\) 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 \( 1 + 9.26T + 27T^{2} \)
5 \( 1 - 16.4T + 125T^{2} \)
7 \( 1 + 34.5T + 343T^{2} \)
11 \( 1 + 7.42T + 1.33e3T^{2} \)
13 \( 1 + 42.0T + 2.19e3T^{2} \)
19 \( 1 + 59.9T + 6.85e3T^{2} \)
23 \( 1 + 49.4T + 1.21e4T^{2} \)
29 \( 1 + 259.T + 2.43e4T^{2} \)
31 \( 1 + 92.2T + 2.97e4T^{2} \)
37 \( 1 - 207.T + 5.06e4T^{2} \)
41 \( 1 + 176.T + 6.89e4T^{2} \)
43 \( 1 - 19.0T + 7.95e4T^{2} \)
47 \( 1 - 80.1T + 1.03e5T^{2} \)
53 \( 1 + 319.T + 1.48e5T^{2} \)
59 \( 1 + 11.0T + 2.05e5T^{2} \)
61 \( 1 + 712.T + 2.26e5T^{2} \)
67 \( 1 - 484.T + 3.00e5T^{2} \)
71 \( 1 + 443.T + 3.57e5T^{2} \)
73 \( 1 - 337.T + 3.89e5T^{2} \)
79 \( 1 + 840.T + 4.93e5T^{2} \)
83 \( 1 + 456.T + 5.71e5T^{2} \)
89 \( 1 + 1.20e3T + 7.04e5T^{2} \)
97 \( 1 + 638.T + 9.12e5T^{2} \)
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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

−9.111052955575728490281696695881, −7.44588448577584684951442963222, −6.76131037574414234334344136971, −6.09512522889573131904731442266, −5.80140960168094151802068854376, −5.01301596038410381174986546693, −3.94753593172332011983694714129, −2.66913963023749185706574898486, −1.64502909447024225429111826256, −0.26065208180831705986839617934, 0.26065208180831705986839617934, 1.64502909447024225429111826256, 2.66913963023749185706574898486, 3.94753593172332011983694714129, 5.01301596038410381174986546693, 5.80140960168094151802068854376, 6.09512522889573131904731442266, 6.76131037574414234334344136971, 7.44588448577584684951442963222, 9.111052955575728490281696695881

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