Properties

Label 2-97461-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s + 13-s − 16-s + 17-s + 8·19-s − 2·20-s − 6·23-s − 25-s − 26-s − 6·29-s + 8·31-s − 5·32-s − 34-s − 4·37-s − 8·38-s + 6·40-s + 6·41-s + 6·46-s + 8·47-s + 50-s − 52-s + 4·53-s + 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s + 0.277·13-s − 1/4·16-s + 0.242·17-s + 1.83·19-s − 0.447·20-s − 1.25·23-s − 1/5·25-s − 0.196·26-s − 1.11·29-s + 1.43·31-s − 0.883·32-s − 0.171·34-s − 0.657·37-s − 1.29·38-s + 0.948·40-s + 0.937·41-s + 0.884·46-s + 1.16·47-s + 0.141·50-s − 0.138·52-s + 0.549·53-s + 0.787·58-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: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.741894579\)
\(L(\frac12)\) \(\approx\) \(1.741894579\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.78663338818051, −13.38299521629038, −12.99390545114819, −12.25019416150775, −11.76002050264505, −11.35699166157548, −10.50002244959829, −10.21614463877282, −9.800884131008007, −9.325919238723366, −9.002530593862608, −8.279240311708217, −7.874213352273381, −7.355751882420938, −6.854158018798325, −5.954650499449272, −5.626040828588740, −5.255353277590044, −4.346264436332511, −3.994882634430669, −3.211607920204152, −2.503533907137763, −1.749113390377450, −1.227820098263625, −0.5150819562350177, 0.5150819562350177, 1.227820098263625, 1.749113390377450, 2.503533907137763, 3.211607920204152, 3.994882634430669, 4.346264436332511, 5.255353277590044, 5.626040828588740, 5.954650499449272, 6.854158018798325, 7.355751882420938, 7.874213352273381, 8.279240311708217, 9.002530593862608, 9.325919238723366, 9.800884131008007, 10.21614463877282, 10.50002244959829, 11.35699166157548, 11.76002050264505, 12.25019416150775, 12.99390545114819, 13.38299521629038, 13.78663338818051

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