Properties

Label 2-97461-1.1-c1-0-14
Degree $2$
Conductor $97461$
Sign $-1$
Analytic cond. $778.230$
Root an. cond. $27.8967$
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  − 2·4-s − 5-s − 6·11-s − 13-s + 4·16-s − 17-s + 7·19-s + 2·20-s + 3·23-s − 4·25-s + 7·29-s + 7·31-s + 8·37-s − 2·41-s − 9·43-s + 12·44-s − 7·47-s + 2·52-s + 3·53-s + 6·55-s − 4·59-s − 6·61-s − 8·64-s + 65-s − 4·67-s + 2·68-s − 12·71-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 1.80·11-s − 0.277·13-s + 16-s − 0.242·17-s + 1.60·19-s + 0.447·20-s + 0.625·23-s − 4/5·25-s + 1.29·29-s + 1.25·31-s + 1.31·37-s − 0.312·41-s − 1.37·43-s + 1.80·44-s − 1.02·47-s + 0.277·52-s + 0.412·53-s + 0.809·55-s − 0.520·59-s − 0.768·61-s − 64-s + 0.124·65-s − 0.488·67-s + 0.242·68-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(778.230\)
Root analytic conductor: \(27.8967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97461} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.89601053063933, −13.44653144022904, −13.19553442566211, −12.70474792924933, −11.96348724298004, −11.73972572913467, −11.12492405153523, −10.35810306972537, −9.995235820551485, −9.807111528614427, −8.970381031734284, −8.487538214979569, −8.045152139054837, −7.549889524178140, −7.275952634050397, −6.237571027267016, −5.851046315499456, −5.011517656409329, −4.854772680159165, −4.406358649957085, −3.416522825480880, −3.060083407203993, −2.532170662935042, −1.452163525239736, −0.6949040149035214, 0, 0.6949040149035214, 1.452163525239736, 2.532170662935042, 3.060083407203993, 3.416522825480880, 4.406358649957085, 4.854772680159165, 5.011517656409329, 5.851046315499456, 6.237571027267016, 7.275952634050397, 7.549889524178140, 8.045152139054837, 8.487538214979569, 8.970381031734284, 9.807111528614427, 9.995235820551485, 10.35810306972537, 11.12492405153523, 11.73972572913467, 11.96348724298004, 12.70474792924933, 13.19553442566211, 13.44653144022904, 13.89601053063933

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