Properties

Label 2-12696-1.1-c1-0-2
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 + 6·11-s − 7·13-s + 2·15-s + 2·17-s + 21-s − 25-s − 27-s + 10·29-s − 8·31-s − 6·33-s + 2·35-s + 7·37-s + 7·39-s − 9·43-s − 2·45-s + 8·47-s − 6·49-s − 2·51-s + 2·53-s − 12·55-s − 4·59-s + 7·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.94·13-s + 0.516·15-s + 0.485·17-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 1.04·33-s + 0.338·35-s + 1.15·37-s + 1.12·39-s − 1.37·43-s − 0.298·45-s + 1.16·47-s − 6/7·49-s − 0.280·51-s + 0.274·53-s − 1.61·55-s − 0.520·59-s + 0.896·61-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

\(L(1)\) \(\approx\) \(0.9757527641\)
\(L(\frac12)\) \(\approx\) \(0.9757527641\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.39752482842099, −15.87637927941690, −14.95922883187414, −14.79153784207370, −14.20161638449090, −13.46949057914357, −12.43076625940836, −12.36424513658527, −11.76373711042092, −11.40162454298065, −10.57386326519033, −9.744877585810196, −9.607612954308954, −8.736484663820790, −7.985436800827872, −7.305088562047469, −6.862385807100753, −6.276322001191068, −5.429533685150459, −4.674428167301910, −4.142274892794457, −3.463625261769833, −2.575571137661976, −1.476126299772459, −0.4780992021249030, 0.4780992021249030, 1.476126299772459, 2.575571137661976, 3.463625261769833, 4.142274892794457, 4.674428167301910, 5.429533685150459, 6.276322001191068, 6.862385807100753, 7.305088562047469, 7.985436800827872, 8.736484663820790, 9.607612954308954, 9.744877585810196, 10.57386326519033, 11.40162454298065, 11.76373711042092, 12.36424513658527, 12.43076625940836, 13.46949057914357, 14.20161638449090, 14.79153784207370, 14.95922883187414, 15.87637927941690, 16.39752482842099

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