Properties

Label 2-97461-1.1-c1-0-13
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  − 2·2-s + 2·4-s − 2·5-s + 4·10-s − 5·11-s + 13-s − 4·16-s − 17-s + 6·19-s − 4·20-s + 10·22-s + 4·23-s − 25-s − 2·26-s − 8·29-s − 31-s + 8·32-s + 2·34-s + 37-s − 12·38-s − 3·41-s + 11·43-s − 10·44-s − 8·46-s + 2·47-s + 2·50-s + 2·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s + 1.26·10-s − 1.50·11-s + 0.277·13-s − 16-s − 0.242·17-s + 1.37·19-s − 0.894·20-s + 2.13·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s − 1.48·29-s − 0.179·31-s + 1.41·32-s + 0.342·34-s + 0.164·37-s − 1.94·38-s − 0.468·41-s + 1.67·43-s − 1.50·44-s − 1.17·46-s + 0.291·47-s + 0.282·50-s + 0.277·52-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 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.84574171743856, −13.56837789207830, −13.04240933935658, −12.44413518711752, −11.94659389845497, −11.22615799529595, −10.99539572203582, −10.69846996467269, −9.937572336813668, −9.523011132529078, −9.097183858401907, −8.526068294721083, −7.936254094776502, −7.605628833984383, −7.396773830579257, −6.728007219467475, −5.945966974540609, −5.192839197877533, −4.973733076436315, −3.968270823426031, −3.596415572857738, −2.696991108175220, −2.266413487790798, −1.339838119096271, −0.6625246472071963, 0, 0.6625246472071963, 1.339838119096271, 2.266413487790798, 2.696991108175220, 3.596415572857738, 3.968270823426031, 4.973733076436315, 5.192839197877533, 5.945966974540609, 6.728007219467475, 7.396773830579257, 7.605628833984383, 7.936254094776502, 8.526068294721083, 9.097183858401907, 9.523011132529078, 9.937572336813668, 10.69846996467269, 10.99539572203582, 11.22615799529595, 11.94659389845497, 12.44413518711752, 13.04240933935658, 13.56837789207830, 13.84574171743856

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