Properties

Label 2-12696-1.1-c1-0-5
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 + 2·11-s − 2·13-s − 2·15-s + 4·17-s + 2·21-s − 25-s − 27-s − 10·29-s − 2·33-s − 4·35-s + 4·37-s + 2·39-s − 6·41-s + 4·43-s + 2·45-s + 8·47-s − 3·49-s − 4·51-s + 6·53-s + 4·55-s + 4·59-s − 8·61-s − 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.516·15-s + 0.970·17-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.348·33-s − 0.676·35-s + 0.657·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s + 0.520·59-s − 1.02·61-s − 0.251·63-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

\(L(1)\) \(\approx\) \(1.694360860\)
\(L(\frac12)\) \(\approx\) \(1.694360860\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−16.66892947654637, −15.76269918242162, −15.19470448992994, −14.54691676803208, −14.04683696870855, −13.36433203836048, −12.91699374425270, −12.31711722003808, −11.80938647720788, −11.14958253931983, −10.44265221013256, −9.868642858117797, −9.484855887361669, −9.006528020103539, −7.964774070716668, −7.319732663475231, −6.731122318199686, −5.980559981313298, −5.671105459503917, −4.965762400146396, −3.996969463840490, −3.401920534622785, −2.394402846776170, −1.649150953825778, −0.6083816606925342, 0.6083816606925342, 1.649150953825778, 2.394402846776170, 3.401920534622785, 3.996969463840490, 4.965762400146396, 5.671105459503917, 5.980559981313298, 6.731122318199686, 7.319732663475231, 7.964774070716668, 9.006528020103539, 9.484855887361669, 9.868642858117797, 10.44265221013256, 11.14958253931983, 11.80938647720788, 12.31711722003808, 12.91699374425270, 13.36433203836048, 14.04683696870855, 14.54691676803208, 15.19470448992994, 15.76269918242162, 16.66892947654637

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