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Properties

Label 2-97461-1.1-c1-0-8
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 − 4·19-s + 4·20-s + 6·22-s + 11·25-s − 26-s + 6·29-s − 10·31-s − 5·32-s − 34-s − 4·37-s + 4·38-s − 12·40-s + 12·43-s + 6·44-s + 8·47-s − 11·50-s − 52-s − 2·53-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 − 0.917·19-s + 0.894·20-s + 1.27·22-s + 11/5·25-s − 0.196·26-s + 1.11·29-s − 1.79·31-s − 0.883·32-s − 0.171·34-s − 0.657·37-s + 0.648·38-s − 1.89·40-s + 1.82·43-s + 0.904·44-s + 1.16·47-s − 1.55·50-s − 0.138·52-s − 0.274·53-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.09643191458571, −13.45218525899803, −12.89499698320301, −12.48883182444074, −12.28741941555273, −11.31732497816921, −10.95640964837231, −10.64339820942912, −10.22493611605136, −9.500918381591383, −8.820533380599733, −8.531662876831352, −8.005522260941269, −7.714885960016704, −7.224358393550991, −6.753348299980418, −5.653844779331029, −5.313454666757787, −4.620268306268158, −4.119584298004523, −3.752192932785271, −2.929582698797712, −2.380518035632467, −1.338937336405626, −0.5117410440012735, 0, 0.5117410440012735, 1.338937336405626, 2.380518035632467, 2.929582698797712, 3.752192932785271, 4.119584298004523, 4.620268306268158, 5.313454666757787, 5.653844779331029, 6.753348299980418, 7.224358393550991, 7.714885960016704, 8.005522260941269, 8.531662876831352, 8.820533380599733, 9.500918381591383, 10.22493611605136, 10.64339820942912, 10.95640964837231, 11.31732497816921, 12.28741941555273, 12.48883182444074, 12.89499698320301, 13.45218525899803, 14.09643191458571

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