Properties

Label 2-12696-1.1-c1-0-17
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 + 2·7-s + 9-s + 2·13-s − 2·15-s − 6·17-s + 2·19-s − 2·21-s − 25-s − 27-s − 6·29-s + 4·35-s + 4·37-s − 2·39-s + 2·41-s − 6·43-s + 2·45-s − 4·47-s − 3·49-s + 6·51-s + 6·53-s − 2·57-s − 4·61-s + 2·63-s + 4·65-s − 10·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.676·35-s + 0.657·37-s − 0.320·39-s + 0.312·41-s − 0.914·43-s + 0.298·45-s − 0.583·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s − 0.512·61-s + 0.251·63-s + 0.496·65-s − 1.22·67-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.63924216711514, −16.08783139007999, −15.44202316419976, −14.91720327818573, −14.29572298657197, −13.62096496629308, −13.22275174810901, −12.79178109165985, −11.77785110525820, −11.42951022119086, −10.94717980179883, −10.29344811308604, −9.678699640468210, −9.070112578400939, −8.491004722925879, −7.717913730550358, −7.052165889605339, −6.330280413130660, −5.834880412856743, −5.205390335903918, −4.533501391432350, −3.869331822728176, −2.771459661327477, −1.888524572233383, −1.355027243707293, 0, 1.355027243707293, 1.888524572233383, 2.771459661327477, 3.869331822728176, 4.533501391432350, 5.205390335903918, 5.834880412856743, 6.330280413130660, 7.052165889605339, 7.717913730550358, 8.491004722925879, 9.070112578400939, 9.678699640468210, 10.29344811308604, 10.94717980179883, 11.42951022119086, 11.77785110525820, 12.79178109165985, 13.22275174810901, 13.62096496629308, 14.29572298657197, 14.91720327818573, 15.44202316419976, 16.08783139007999, 16.63924216711514

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