Properties

Label 2-2312-1.1-c3-0-136
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.0·3-s + 13.8·5-s − 18.1·7-s + 73.9·9-s + 46.2·11-s + 65.8·13-s − 139.·15-s − 19.0·19-s + 181.·21-s − 194.·23-s + 66.7·25-s − 471.·27-s + 118.·29-s − 4.17·31-s − 465.·33-s − 250.·35-s − 270.·37-s − 661.·39-s + 31.6·41-s + 9.51·43-s + 1.02e3·45-s − 471.·47-s − 15.0·49-s − 96.9·53-s + 641.·55-s + 191.·57-s + 854.·59-s + ⋯
L(s)  = 1  − 1.93·3-s + 1.23·5-s − 0.977·7-s + 2.73·9-s + 1.26·11-s + 1.40·13-s − 2.39·15-s − 0.230·19-s + 1.89·21-s − 1.76·23-s + 0.534·25-s − 3.35·27-s + 0.759·29-s − 0.0241·31-s − 2.45·33-s − 1.21·35-s − 1.20·37-s − 2.71·39-s + 0.120·41-s + 0.0337·43-s + 3.39·45-s − 1.46·47-s − 0.0439·49-s − 0.251·53-s + 1.57·55-s + 0.444·57-s + 1.88·59-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: \(1\)
Selberg data: \((2,\ 2312,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 10.0T + 27T^{2} \)
5 \( 1 - 13.8T + 125T^{2} \)
7 \( 1 + 18.1T + 343T^{2} \)
11 \( 1 - 46.2T + 1.33e3T^{2} \)
13 \( 1 - 65.8T + 2.19e3T^{2} \)
19 \( 1 + 19.0T + 6.85e3T^{2} \)
23 \( 1 + 194.T + 1.21e4T^{2} \)
29 \( 1 - 118.T + 2.43e4T^{2} \)
31 \( 1 + 4.17T + 2.97e4T^{2} \)
37 \( 1 + 270.T + 5.06e4T^{2} \)
41 \( 1 - 31.6T + 6.89e4T^{2} \)
43 \( 1 - 9.51T + 7.95e4T^{2} \)
47 \( 1 + 471.T + 1.03e5T^{2} \)
53 \( 1 + 96.9T + 1.48e5T^{2} \)
59 \( 1 - 854.T + 2.05e5T^{2} \)
61 \( 1 - 171.T + 2.26e5T^{2} \)
67 \( 1 + 8.33T + 3.00e5T^{2} \)
71 \( 1 + 51.0T + 3.57e5T^{2} \)
73 \( 1 + 946.T + 3.89e5T^{2} \)
79 \( 1 + 16.0T + 4.93e5T^{2} \)
83 \( 1 - 1.24e3T + 5.71e5T^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 1.16e3T + 9.12e5T^{2} \)
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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

−8.348752472078381031637939738785, −6.82876636275730201103002915091, −6.52380563790650395256302617522, −5.98167716574475503512952756192, −5.49052749462300012283644548757, −4.32751388823861433388128374344, −3.60073927284605671710273959125, −1.84375225055718099348339490184, −1.15440612194477680239692256459, 0, 1.15440612194477680239692256459, 1.84375225055718099348339490184, 3.60073927284605671710273959125, 4.32751388823861433388128374344, 5.49052749462300012283644548757, 5.98167716574475503512952756192, 6.52380563790650395256302617522, 6.82876636275730201103002915091, 8.348752472078381031637939738785

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