Properties

Label 2-12696-1.1-c1-0-16
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 − 7-s + 9-s + 2·11-s + 13-s − 2·15-s − 2·17-s + 21-s − 25-s − 27-s − 2·29-s − 2·33-s − 2·35-s − 37-s − 39-s − 43-s + 2·45-s + 4·47-s − 6·49-s + 2·51-s − 6·53-s + 4·55-s + 8·59-s − 61-s − 63-s + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s − 0.485·17-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.348·33-s − 0.338·35-s − 0.164·37-s − 0.160·39-s − 0.152·43-s + 0.298·45-s + 0.583·47-s − 6/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s + 1.04·59-s − 0.128·61-s − 0.125·63-s + 0.248·65-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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.67633220627633, −15.95855490402437, −15.69220868430928, −14.75570389430245, −14.37729390535816, −13.57608571828433, −13.25400301470094, −12.66643932893346, −11.98426605422650, −11.43943551289108, −10.84494641116832, −10.22272705031955, −9.643270373566037, −9.204707187588461, −8.512932533932064, −7.708001425380414, −6.852744080808627, −6.482660706649328, −5.810462947992435, −5.328766240172171, −4.434148614187370, −3.793642873371797, −2.868743874732161, −1.954686748393667, −1.239574416983083, 0, 1.239574416983083, 1.954686748393667, 2.868743874732161, 3.793642873371797, 4.434148614187370, 5.328766240172171, 5.810462947992435, 6.482660706649328, 6.852744080808627, 7.708001425380414, 8.512932533932064, 9.204707187588461, 9.643270373566037, 10.22272705031955, 10.84494641116832, 11.43943551289108, 11.98426605422650, 12.66643932893346, 13.25400301470094, 13.57608571828433, 14.37729390535816, 14.75570389430245, 15.69220868430928, 15.95855490402437, 16.67633220627633

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