Properties

Label 2-8280-1.1-c1-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3.12·7-s + 4·11-s + 3.56·13-s − 5.12·17-s + 4·19-s − 23-s + 25-s + 4.43·29-s + 5.56·31-s + 3.12·35-s + 1.12·37-s + 3.56·41-s − 0.876·43-s − 8.68·47-s + 2.75·49-s − 12.2·53-s − 4·55-s − 10.2·59-s + 2.87·61-s − 3.56·65-s − 10.2·67-s + 8.68·71-s + 12.4·73-s − 12.4·77-s + 6.24·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.18·7-s + 1.20·11-s + 0.987·13-s − 1.24·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s + 0.824·29-s + 0.998·31-s + 0.527·35-s + 0.184·37-s + 0.556·41-s − 0.133·43-s − 1.26·47-s + 0.393·49-s − 1.68·53-s − 0.539·55-s − 1.33·59-s + 0.368·61-s − 0.441·65-s − 1.25·67-s + 1.03·71-s + 1.45·73-s − 1.42·77-s + 0.702·79-s − 1.31·83-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: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600087987\)
\(L(\frac12)\) \(\approx\) \(1.600087987\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 - 4.43T + 29T^{2} \)
31 \( 1 - 5.56T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 + 0.876T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 8.68T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 6.24T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 0.246T + 97T^{2} \)
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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

−7.84707399761984396157813710519, −6.90898529115688709566366017144, −6.39337597931529919222221143614, −6.10199287219393165201701679281, −4.84551047265947340719162026368, −4.20855897884656671245321242527, −3.43834793725010218979562495934, −2.92906737971132218908179495127, −1.64704929413425993711186478908, −0.63811701482272834039898789466, 0.63811701482272834039898789466, 1.64704929413425993711186478908, 2.92906737971132218908179495127, 3.43834793725010218979562495934, 4.20855897884656671245321242527, 4.84551047265947340719162026368, 6.10199287219393165201701679281, 6.39337597931529919222221143614, 6.90898529115688709566366017144, 7.84707399761984396157813710519

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