Properties

Label 2-12696-1.1-c1-0-14
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 2·17-s + 4·19-s − 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s + 2·53-s − 8·55-s − 4·57-s + 4·59-s + 2·61-s − 4·65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s + 0.274·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-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: \(1\)
Selberg data: \((2,\ 12696,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
23 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−16.58593172334788, −16.00601619749672, −15.51806779585239, −15.06094949899946, −14.05870319008221, −13.74977549082251, −13.26560976153551, −12.59945965060409, −12.00035150878379, −11.52261679664918, −10.68353274937486, −10.22172520089540, −9.821544784635579, −9.217247137528152, −8.203132085402416, −7.932741438944089, −6.810918917143638, −6.618948213116475, −5.686930301735402, −5.086374372400919, −4.829312432515918, −3.665115293059204, −2.741283731455776, −2.155783072374226, −1.138507899479109, 0, 1.138507899479109, 2.155783072374226, 2.741283731455776, 3.665115293059204, 4.829312432515918, 5.086374372400919, 5.686930301735402, 6.618948213116475, 6.810918917143638, 7.932741438944089, 8.203132085402416, 9.217247137528152, 9.821544784635579, 10.22172520089540, 10.68353274937486, 11.52261679664918, 12.00035150878379, 12.59945965060409, 13.26560976153551, 13.74977549082251, 14.05870319008221, 15.06094949899946, 15.51806779585239, 16.00601619749672, 16.58593172334788

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