Properties

Label 2-12696-1.1-c1-0-18
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 4·7-s + 9-s − 13-s + 3·15-s − 2·17-s + 4·21-s + 4·25-s − 27-s − 29-s + 8·31-s + 12·35-s − 6·37-s + 39-s − 5·41-s − 12·43-s − 3·45-s − 12·47-s + 9·49-s + 2·51-s + 9·53-s − 8·59-s − 11·61-s − 4·63-s + 3·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.774·15-s − 0.485·17-s + 0.872·21-s + 4/5·25-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 2.02·35-s − 0.986·37-s + 0.160·39-s − 0.780·41-s − 1.82·43-s − 0.447·45-s − 1.75·47-s + 9/7·49-s + 0.280·51-s + 1.23·53-s − 1.04·59-s − 1.40·61-s − 0.503·63-s + 0.372·65-s + 1.46·67-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: \(2\)
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 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−16.84519915375185, −16.21564863414140, −15.66106365037833, −15.47118494007411, −14.85286764428755, −13.93884137850783, −13.25484233437250, −12.89780159904183, −12.11552473003630, −11.85671545447825, −11.31164831378004, −10.49183964811628, −10.01791354623845, −9.478585335557876, −8.566428290642785, −8.168860647671249, −7.240559419841085, −6.798309323613889, −6.355123275103673, −5.464281574515247, −4.683637763754160, −4.062991579797594, −3.333495422473145, −2.818240507415747, −1.455486754786297, 0, 0, 1.455486754786297, 2.818240507415747, 3.333495422473145, 4.062991579797594, 4.683637763754160, 5.464281574515247, 6.355123275103673, 6.798309323613889, 7.240559419841085, 8.168860647671249, 8.566428290642785, 9.478585335557876, 10.01791354623845, 10.49183964811628, 11.31164831378004, 11.85671545447825, 12.11552473003630, 12.89780159904183, 13.25484233437250, 13.93884137850783, 14.85286764428755, 15.47118494007411, 15.66106365037833, 16.21564863414140, 16.84519915375185

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