Properties

Label 2-12696-1.1-c1-0-13
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

−16.42994405447312, −16.09124013161308, −15.50763118560653, −14.96604888798346, −14.48194442781752, −13.62297153118489, −13.18313135849600, −12.43702749440930, −12.00438460017363, −11.38183581499391, −11.03566743007182, −10.29683587809196, −9.824869239328240, −8.902609561274123, −8.339154763367204, −7.621087823946684, −7.352148113091306, −6.440773120963800, −5.725844157769832, −5.169827542517263, −4.402911119017581, −3.824277992594454, −3.023711053660488, −2.020823765864921, −0.9944734700913198, 0, 0.9944734700913198, 2.020823765864921, 3.023711053660488, 3.824277992594454, 4.402911119017581, 5.169827542517263, 5.725844157769832, 6.440773120963800, 7.352148113091306, 7.621087823946684, 8.339154763367204, 8.902609561274123, 9.824869239328240, 10.29683587809196, 11.03566743007182, 11.38183581499391, 12.00438460017363, 12.43702749440930, 13.18313135849600, 13.62297153118489, 14.48194442781752, 14.96604888798346, 15.50763118560653, 16.09124013161308, 16.42994405447312

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