Properties

Label 2-12696-1.1-c1-0-4
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 − 4·5-s − 2·7-s + 9-s + 2·13-s + 4·15-s + 4·17-s + 6·19-s + 2·21-s + 11·25-s − 27-s + 10·29-s + 4·31-s + 8·35-s + 2·37-s − 2·39-s − 6·41-s + 6·43-s − 4·45-s + 8·47-s − 3·49-s − 4·51-s − 8·53-s − 6·57-s − 4·59-s + 2·61-s − 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s + 0.970·17-s + 1.37·19-s + 0.436·21-s + 11/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.914·43-s − 0.596·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s − 1.09·53-s − 0.794·57-s − 0.520·59-s + 0.256·61-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

\(L(1)\) \(\approx\) \(1.086140342\)
\(L(\frac12)\) \(\approx\) \(1.086140342\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−16.01050293422491, −15.77880852555966, −15.62961962446685, −14.62576601879977, −14.10846946622058, −13.39986884975460, −12.63605442601823, −12.21101807672371, −11.71627504146945, −11.41703361911143, −10.47839158590016, −10.19182897851119, −9.309129197675084, −8.644829149168565, −7.882137790808524, −7.571995508768428, −6.811647322136453, −6.252676616614727, −5.444942112768036, −4.654953159294805, −4.111337228489088, −3.205850335839674, −3.013018497094497, −1.208699027395361, −0.5727725240081691, 0.5727725240081691, 1.208699027395361, 3.013018497094497, 3.205850335839674, 4.111337228489088, 4.654953159294805, 5.444942112768036, 6.252676616614727, 6.811647322136453, 7.571995508768428, 7.882137790808524, 8.644829149168565, 9.309129197675084, 10.19182897851119, 10.47839158590016, 11.41703361911143, 11.71627504146945, 12.21101807672371, 12.63605442601823, 13.39986884975460, 14.10846946622058, 14.62576601879977, 15.62961962446685, 15.77880852555966, 16.01050293422491

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