Properties

Label 2-12696-1.1-c1-0-8
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 + 7-s + 9-s + 5·11-s + 4·13-s − 2·15-s − 2·17-s − 21-s − 25-s − 27-s − 29-s + 3·31-s − 5·33-s + 2·35-s + 4·37-s − 4·39-s − 2·43-s + 2·45-s + 8·47-s − 6·49-s + 2·51-s + 9·53-s + 10·55-s + 7·59-s + 4·61-s + 63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s − 0.516·15-s − 0.485·17-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 0.538·31-s − 0.870·33-s + 0.338·35-s + 0.657·37-s − 0.640·39-s − 0.304·43-s + 0.298·45-s + 1.16·47-s − 6/7·49-s + 0.280·51-s + 1.23·53-s + 1.34·55-s + 0.911·59-s + 0.512·61-s + 0.125·63-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

\(L(1)\) \(\approx\) \(2.833109729\)
\(L(\frac12)\) \(\approx\) \(2.833109729\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−16.31034807325036, −15.92995392768628, −15.02182116515789, −14.66147283549408, −13.95391028670079, −13.39419936958970, −13.13435870457426, −12.04461182149160, −11.82190776666198, −11.13497366313442, −10.63871466078352, −9.928085466447988, −9.360773641906249, −8.804231395116648, −8.233377211300098, −7.240398434963280, −6.679479796421248, −5.982612574921075, −5.791461877646404, −4.748658304972288, −4.147464993010050, −3.463004611790009, −2.283404228804247, −1.548268233705654, −0.8565383517966445, 0.8565383517966445, 1.548268233705654, 2.283404228804247, 3.463004611790009, 4.147464993010050, 4.748658304972288, 5.791461877646404, 5.982612574921075, 6.679479796421248, 7.240398434963280, 8.233377211300098, 8.804231395116648, 9.360773641906249, 9.928085466447988, 10.63871466078352, 11.13497366313442, 11.82190776666198, 12.04461182149160, 13.13435870457426, 13.39419936958970, 13.95391028670079, 14.66147283549408, 15.02182116515789, 15.92995392768628, 16.31034807325036

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