Properties

Label 2-12696-1.1-c1-0-12
Degree $2$
Conductor $12696$
Sign $1$
Analytic cond. $101.378$
Root an. cond. $10.0686$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 2·7-s + 9-s + 4·11-s − 6·13-s + 2·15-s + 6·17-s + 6·19-s + 2·21-s − 25-s + 27-s − 6·29-s + 8·31-s + 4·33-s + 4·35-s + 8·37-s − 6·39-s − 6·41-s − 10·43-s + 2·45-s + 4·47-s − 3·49-s + 6·51-s − 2·53-s + 8·55-s + 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.516·15-s + 1.45·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 1.31·37-s − 0.960·39-s − 0.937·41-s − 1.52·43-s + 0.298·45-s + 0.583·47-s − 3/7·49-s + 0.840·51-s − 0.274·53-s + 1.07·55-s + 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12696\)    =    \(2^{3} \cdot 3 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(101.378\)
Root analytic conductor: \(10.0686\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12696} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12696,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.285824650\)
\(L(\frac12)\) \(\approx\) \(4.285824650\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.59533518287548, −15.61762927673653, −14.81019541516849, −14.55489111660723, −14.25667173586826, −13.53470247139311, −13.10258522857955, −12.04062960129568, −11.92576591936095, −11.33789062719572, −10.10463926418835, −9.918708214200294, −9.504859783993659, −8.825180599265227, −7.894247331557890, −7.657274134965222, −6.865414976922011, −6.165344676683126, −5.246400042389670, −5.009799029245355, −3.989135278291676, −3.252879146249315, −2.434832528188577, −1.693327154735086, −0.9790974914612601, 0.9790974914612601, 1.693327154735086, 2.434832528188577, 3.252879146249315, 3.989135278291676, 5.009799029245355, 5.246400042389670, 6.165344676683126, 6.865414976922011, 7.657274134965222, 7.894247331557890, 8.825180599265227, 9.504859783993659, 9.918708214200294, 10.10463926418835, 11.33789062719572, 11.92576591936095, 12.04062960129568, 13.10258522857955, 13.53470247139311, 14.25667173586826, 14.55489111660723, 14.81019541516849, 15.61762927673653, 16.59533518287548

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