Properties

Label 2-8280-1.1-c1-0-102
Degree $2$
Conductor $8280$
Sign $-1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4.19·7-s − 1.38·11-s − 6.52·13-s − 5.47·17-s − 3.72·19-s + 23-s + 25-s + 0.0800·29-s + 6.53·31-s + 4.19·35-s − 4.51·37-s + 8.32·41-s + 9.02·43-s + 0.818·47-s + 10.5·49-s − 9.55·53-s − 1.38·55-s − 1.34·59-s − 12.3·61-s − 6.52·65-s + 0.926·67-s − 13.6·71-s + 2.65·73-s − 5.82·77-s + 10.6·79-s − 10.2·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.58·7-s − 0.418·11-s − 1.80·13-s − 1.32·17-s − 0.853·19-s + 0.208·23-s + 0.200·25-s + 0.0148·29-s + 1.17·31-s + 0.709·35-s − 0.742·37-s + 1.29·41-s + 1.37·43-s + 0.119·47-s + 1.51·49-s − 1.31·53-s − 0.187·55-s − 0.174·59-s − 1.58·61-s − 0.808·65-s + 0.113·67-s − 1.62·71-s + 0.311·73-s − 0.664·77-s + 1.19·79-s − 1.12·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 - 4.19T + 7T^{2} \)
11 \( 1 + 1.38T + 11T^{2} \)
13 \( 1 + 6.52T + 13T^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
29 \( 1 - 0.0800T + 29T^{2} \)
31 \( 1 - 6.53T + 31T^{2} \)
37 \( 1 + 4.51T + 37T^{2} \)
41 \( 1 - 8.32T + 41T^{2} \)
43 \( 1 - 9.02T + 43T^{2} \)
47 \( 1 - 0.818T + 47T^{2} \)
53 \( 1 + 9.55T + 53T^{2} \)
59 \( 1 + 1.34T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 0.926T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 2.65T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 5.66T + 89T^{2} \)
97 \( 1 + 9.57T + 97T^{2} \)
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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

−7.59069619131792217100164448383, −6.84663819119244806080461874146, −6.05137416558126307264161702933, −5.20201221186780213169053691865, −4.62426348158729620846444202213, −4.29993938978766682098919720486, −2.67986362879437695633631776614, −2.32216473244915260411562862759, −1.42093274705204756544566029601, 0, 1.42093274705204756544566029601, 2.32216473244915260411562862759, 2.67986362879437695633631776614, 4.29993938978766682098919720486, 4.62426348158729620846444202213, 5.20201221186780213169053691865, 6.05137416558126307264161702933, 6.84663819119244806080461874146, 7.59069619131792217100164448383

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