Properties

Label 2-8280-1.1-c1-0-26
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 − 4·11-s + 2·13-s − 2·17-s + 23-s + 25-s + 6·29-s − 6·37-s − 2·41-s + 12·43-s − 7·49-s − 10·53-s − 4·55-s + 12·59-s − 6·61-s + 2·65-s + 12·67-s + 12·71-s + 2·73-s + 16·79-s − 16·83-s − 2·85-s − 14·89-s + 6·97-s − 10·101-s + 8·103-s + 10·109-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.208·23-s + 1/5·25-s + 1.11·29-s − 0.986·37-s − 0.312·41-s + 1.82·43-s − 49-s − 1.37·53-s − 0.539·55-s + 1.56·59-s − 0.768·61-s + 0.248·65-s + 1.46·67-s + 1.42·71-s + 0.234·73-s + 1.80·79-s − 1.75·83-s − 0.216·85-s − 1.48·89-s + 0.609·97-s − 0.995·101-s + 0.788·103-s + 0.957·109-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

\(L(1)\) \(\approx\) \(1.936114224\)
\(L(\frac12)\) \(\approx\) \(1.936114224\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.960972345802583748236505828552, −7.01084779076761750159357176047, −6.47911974515012984213798613717, −5.66837052279709760331333880697, −5.09224079361795035060815253989, −4.37260352652791807998889734777, −3.38434121556416518004393820067, −2.63608801407211173598861341226, −1.85232171975255725425548767101, −0.67515771210227546855419498872, 0.67515771210227546855419498872, 1.85232171975255725425548767101, 2.63608801407211173598861341226, 3.38434121556416518004393820067, 4.37260352652791807998889734777, 5.09224079361795035060815253989, 5.66837052279709760331333880697, 6.47911974515012984213798613717, 7.01084779076761750159357176047, 7.960972345802583748236505828552

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