Properties

Label 2-8280-1.1-c1-0-12
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
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  − 5-s − 3·7-s + 4·11-s − 3·17-s − 4·19-s − 23-s + 25-s − 29-s + 31-s + 3·35-s + 37-s − 3·41-s − 12·43-s + 10·47-s + 2·49-s + 9·53-s − 4·55-s + 9·59-s − 10·61-s + 67-s − 5·71-s − 12·77-s + 4·79-s + 9·83-s + 3·85-s + 8·89-s + 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 1.20·11-s − 0.727·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.185·29-s + 0.179·31-s + 0.507·35-s + 0.164·37-s − 0.468·41-s − 1.82·43-s + 1.45·47-s + 2/7·49-s + 1.23·53-s − 0.539·55-s + 1.17·59-s − 1.28·61-s + 0.122·67-s − 0.593·71-s − 1.36·77-s + 0.450·79-s + 0.987·83-s + 0.325·85-s + 0.847·89-s + 0.410·95-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: yes
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.174184319\)
\(L(\frac12)\) \(\approx\) \(1.174184319\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.76364132546118976223330171554, −6.89169151718913384751733399228, −6.56219107141053445824983984564, −5.94697678821822871064527639127, −4.92244943626343921597989137252, −4.06056711558805015424167148621, −3.66295621882111115467321009261, −2.73915226491007000399655434742, −1.77746755409833811183434449578, −0.52204454276184848581613183790, 0.52204454276184848581613183790, 1.77746755409833811183434449578, 2.73915226491007000399655434742, 3.66295621882111115467321009261, 4.06056711558805015424167148621, 4.92244943626343921597989137252, 5.94697678821822871064527639127, 6.56219107141053445824983984564, 6.89169151718913384751733399228, 7.76364132546118976223330171554

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