Properties

Label 2-12696-1.1-c1-0-19
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 2·7-s + 9-s − 4·11-s − 6·13-s − 2·15-s − 6·17-s − 6·19-s − 2·21-s − 25-s + 27-s − 6·29-s + 8·31-s − 4·33-s + 4·35-s − 8·37-s − 6·39-s − 6·41-s + 10·43-s − 2·45-s + 4·47-s − 3·49-s − 6·51-s + 2·53-s + 8·55-s − 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.516·15-s − 1.45·17-s − 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s − 1.31·37-s − 0.960·39-s − 0.937·41-s + 1.52·43-s − 0.298·45-s + 0.583·47-s − 3/7·49-s − 0.840·51-s + 0.274·53-s + 1.07·55-s − 0.794·57-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−16.92320699149322, −16.00228115278920, −15.65159011345900, −15.25660120716773, −14.80581771544613, −14.03836258492714, −13.30383507961763, −12.97342054107590, −12.39884224249107, −11.84799633606620, −11.11714990417846, −10.34975668416641, −10.10328423391297, −9.238521191610947, −8.660759052745661, −8.135676176311952, −7.297499107964155, −7.156046382938926, −6.246876883297156, −5.405381936619046, −4.406178331374931, −4.303602692612675, −3.156196579875428, −2.591440933046888, −1.968159665161164, 0, 0, 1.968159665161164, 2.591440933046888, 3.156196579875428, 4.303602692612675, 4.406178331374931, 5.405381936619046, 6.246876883297156, 7.156046382938926, 7.297499107964155, 8.135676176311952, 8.660759052745661, 9.238521191610947, 10.10328423391297, 10.34975668416641, 11.11714990417846, 11.84799633606620, 12.39884224249107, 12.97342054107590, 13.30383507961763, 14.03836258492714, 14.80581771544613, 15.25660120716773, 15.65159011345900, 16.00228115278920, 16.92320699149322

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