Properties

Label 2-75712-1.1-c1-0-36
Degree $2$
Conductor $75712$
Sign $-1$
Analytic cond. $604.563$
Root an. cond. $24.5878$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 7-s + 9-s + 2·11-s + 2·15-s + 17-s − 4·19-s − 2·21-s − 2·23-s − 4·25-s + 4·27-s + 5·29-s − 6·31-s − 4·33-s − 35-s − 7·37-s − 7·41-s − 2·43-s − 45-s + 49-s − 2·51-s + 9·53-s − 2·55-s + 8·57-s − 6·59-s − 5·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.516·15-s + 0.242·17-s − 0.917·19-s − 0.436·21-s − 0.417·23-s − 4/5·25-s + 0.769·27-s + 0.928·29-s − 1.07·31-s − 0.696·33-s − 0.169·35-s − 1.15·37-s − 1.09·41-s − 0.304·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 1.23·53-s − 0.269·55-s + 1.05·57-s − 0.781·59-s − 0.640·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75712\)    =    \(2^{6} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(604.563\)
Root analytic conductor: \(24.5878\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.25823065768273, −13.89870564024624, −13.23184200623251, −12.61394053989635, −12.11389701162362, −11.81233624911565, −11.44997100691112, −10.85789331276435, −10.32224285270334, −10.10741234981176, −9.123010891376088, −8.747462667708451, −8.210754860264469, −7.583145888602899, −7.025627999180221, −6.460286210390413, −6.038909037573363, −5.455238774349370, −4.889505761111760, −4.405616371962461, −3.734858953394550, −3.221151339311812, −2.156352929968410, −1.623437969711604, −0.7026921560807695, 0, 0.7026921560807695, 1.623437969711604, 2.156352929968410, 3.221151339311812, 3.734858953394550, 4.405616371962461, 4.889505761111760, 5.455238774349370, 6.038909037573363, 6.460286210390413, 7.025627999180221, 7.583145888602899, 8.210754860264469, 8.747462667708451, 9.123010891376088, 10.10741234981176, 10.32224285270334, 10.85789331276435, 11.44997100691112, 11.81233624911565, 12.11389701162362, 12.61394053989635, 13.23184200623251, 13.89870564024624, 14.25823065768273

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