Properties

Label 2-12696-1.1-c1-0-15
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 + 4·7-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 2·17-s − 4·21-s − 25-s − 27-s − 2·29-s − 4·33-s − 8·35-s + 10·37-s + 2·39-s − 6·41-s − 8·43-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s − 8·55-s − 4·59-s − 14·61-s + 4·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 1.35·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.07·55-s − 0.520·59-s − 1.79·61-s + 0.503·63-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: \(1\)
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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.53861676036145, −16.23505810864679, −15.20323528231294, −14.87496375917443, −14.61774293357148, −13.78860887294808, −13.19996699976314, −12.26743356831438, −11.80452128059251, −11.63671237363940, −11.06253535819688, −10.37105612785704, −9.671261423732871, −8.975498681721978, −8.252711703295064, −7.743159706856119, −7.279424752641452, −6.480772092288813, −5.776890292134297, −4.967753347802210, −4.478665094869005, −3.930569908013865, −3.028031709132341, −1.779060228872881, −1.239013225900669, 0, 1.239013225900669, 1.779060228872881, 3.028031709132341, 3.930569908013865, 4.478665094869005, 4.967753347802210, 5.776890292134297, 6.480772092288813, 7.279424752641452, 7.743159706856119, 8.252711703295064, 8.975498681721978, 9.671261423732871, 10.37105612785704, 11.06253535819688, 11.63671237363940, 11.80452128059251, 12.26743356831438, 13.19996699976314, 13.78860887294808, 14.61774293357148, 14.87496375917443, 15.20323528231294, 16.23505810864679, 16.53861676036145

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