Properties

Label 2-97461-1.1-c1-0-22
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-s − 4-s + 4·5-s − 3·8-s + 4·10-s − 6·11-s + 13-s − 16-s + 17-s − 8·19-s − 4·20-s − 6·22-s − 4·23-s + 11·25-s + 26-s + 6·29-s + 2·31-s + 5·32-s + 34-s − 8·37-s − 8·38-s − 12·40-s + 4·43-s + 6·44-s − 4·46-s + 11·50-s − 52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s + 1.26·10-s − 1.80·11-s + 0.277·13-s − 1/4·16-s + 0.242·17-s − 1.83·19-s − 0.894·20-s − 1.27·22-s − 0.834·23-s + 11/5·25-s + 0.196·26-s + 1.11·29-s + 0.359·31-s + 0.883·32-s + 0.171·34-s − 1.31·37-s − 1.29·38-s − 1.89·40-s + 0.609·43-s + 0.904·44-s − 0.589·46-s + 1.55·50-s − 0.138·52-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 \( 1 - T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.90394428205191, −13.55190634976511, −13.05981780791208, −12.82689765396675, −12.38305217593495, −11.75440504651341, −10.87850092518407, −10.39368964225748, −10.16134759490114, −9.797682823924777, −8.936156455903537, −8.605029256346144, −8.271219813336548, −7.422161985037742, −6.706363373764989, −6.066229289761516, −5.913584377017336, −5.311454426225033, −4.825219929105924, −4.406005226273471, −3.549850886197715, −2.879220975970073, −2.311227953967161, −1.981028669205858, −0.8852702126675879, 0, 0.8852702126675879, 1.981028669205858, 2.311227953967161, 2.879220975970073, 3.549850886197715, 4.406005226273471, 4.825219929105924, 5.311454426225033, 5.913584377017336, 6.066229289761516, 6.706363373764989, 7.422161985037742, 8.271219813336548, 8.605029256346144, 8.936156455903537, 9.797682823924777, 10.16134759490114, 10.39368964225748, 10.87850092518407, 11.75440504651341, 12.38305217593495, 12.82689765396675, 13.05981780791208, 13.55190634976511, 13.90394428205191

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