Properties

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

\(L(1)\) \(\approx\) \(0.6993518724\)
\(L(\frac12)\) \(\approx\) \(0.6993518724\)
\(L(\frac{3}{2})\) 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

−16.17612854742172, −15.64513707063622, −15.58028889862745, −14.74383083403552, −13.88338400763186, −13.41661425825334, −12.79700145637873, −12.33253962339883, −11.72937439193646, −11.01365147638201, −10.77536134826246, −10.02108444385865, −9.481837087085047, −8.402733713617357, −8.198577447810319, −7.416004539482563, −6.905387064662747, −5.969320144088707, −5.600979761115533, −4.777421070062275, −4.068818664824007, −3.373923176965240, −2.666683995916309, −1.480613293026178, −0.3988646116142618, 0.3988646116142618, 1.480613293026178, 2.666683995916309, 3.373923176965240, 4.068818664824007, 4.777421070062275, 5.600979761115533, 5.969320144088707, 6.905387064662747, 7.416004539482563, 8.198577447810319, 8.402733713617357, 9.481837087085047, 10.02108444385865, 10.77536134826246, 11.01365147638201, 11.72937439193646, 12.33253962339883, 12.79700145637873, 13.41661425825334, 13.88338400763186, 14.74383083403552, 15.58028889862745, 15.64513707063622, 16.17612854742172

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