Properties

Label 2-12696-1.1-c1-0-9
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 + 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: \(0\)
Selberg data: \((2,\ 12696,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.152455139\)
\(L(\frac12)\) \(\approx\) \(3.152455139\)
\(L(\frac{3}{2})\) 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

−16.45492424940707, −15.79225134778509, −14.99971339919818, −14.52257837891595, −14.12314951596129, −13.47165221019860, −12.99810199105107, −12.20411624647541, −11.69591564418042, −11.15466079945661, −10.51506378833773, −10.03814771779050, −9.433661402243520, −8.798012180618879, −7.908581502693089, −7.634518829613816, −6.570526691960593, −6.089237530523960, −5.477488471545184, −4.804039873430354, −4.471666981126753, −3.203997168452351, −2.245608807387523, −1.653647879148321, −0.8818282234756999, 0.8818282234756999, 1.653647879148321, 2.245608807387523, 3.203997168452351, 4.471666981126753, 4.804039873430354, 5.477488471545184, 6.089237530523960, 6.570526691960593, 7.634518829613816, 7.908581502693089, 8.798012180618879, 9.433661402243520, 10.03814771779050, 10.51506378833773, 11.15466079945661, 11.69591564418042, 12.20411624647541, 12.99810199105107, 13.47165221019860, 14.12314951596129, 14.52257837891595, 14.99971339919818, 15.79225134778509, 16.45492424940707

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