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Properties

Label 2-12789-1.1-c1-0-4
Degree $2$
Conductor $12789$
Sign $-1$
Analytic cond. $102.120$
Root an. cond. $10.1054$
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  + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 4·11-s + 2·13-s − 16-s + 2·17-s + 4·19-s + 2·20-s − 4·22-s − 25-s + 2·26-s − 29-s + 8·31-s + 5·32-s + 2·34-s − 10·37-s + 4·38-s + 6·40-s − 6·41-s + 12·43-s + 4·44-s − 8·47-s − 50-s − 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.20·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.185·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s − 1.64·37-s + 0.648·38-s + 0.948·40-s − 0.937·41-s + 1.82·43-s + 0.603·44-s − 1.16·47-s − 0.141·50-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12789 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12789\)    =    \(3^{2} \cdot 7^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(102.120\)
Root analytic conductor: \(10.1054\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12789,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 \)
29 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.20913749747659, −15.82280243047844, −15.54159585484132, −14.85334343888596, −14.22185698079179, −13.67792711511348, −13.29327254924692, −12.58583618872979, −12.13706495407205, −11.58590604004209, −11.01363892204498, −10.13670392672195, −9.800837126270721, −8.864769804485816, −8.306450905886112, −7.864756715982596, −7.171614311101287, −6.328229883821062, −5.556395526888723, −5.097865255122100, −4.459211032063406, −3.594564910227314, −3.296480908421373, −2.366359405014978, −0.9785561081899453, 0, 0.9785561081899453, 2.366359405014978, 3.296480908421373, 3.594564910227314, 4.459211032063406, 5.097865255122100, 5.556395526888723, 6.328229883821062, 7.171614311101287, 7.864756715982596, 8.306450905886112, 8.864769804485816, 9.800837126270721, 10.13670392672195, 11.01363892204498, 11.58590604004209, 12.13706495407205, 12.58583618872979, 13.29327254924692, 13.67792711511348, 14.22185698079179, 14.85334343888596, 15.54159585484132, 15.82280243047844, 16.20913749747659

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