Properties

Label 2-8280-1.1-c1-0-21
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 + 3.95·7-s + 0.957·11-s − 2.74·13-s − 5.74·17-s − 6.74·19-s − 23-s + 25-s − 5.21·29-s − 5.95·31-s − 3.95·35-s + 9.12·37-s − 0.252·41-s + 8·43-s − 5.49·47-s + 8.66·49-s + 7.12·53-s − 0.957·55-s + 4.78·59-s + 12.4·61-s + 2.74·65-s + 9.12·67-s − 1.66·71-s + 12.3·73-s + 3.78·77-s + 11.8·79-s − 0.704·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.49·7-s + 0.288·11-s − 0.761·13-s − 1.39·17-s − 1.54·19-s − 0.208·23-s + 0.200·25-s − 0.967·29-s − 1.07·31-s − 0.668·35-s + 1.50·37-s − 0.0394·41-s + 1.21·43-s − 0.801·47-s + 1.23·49-s + 0.978·53-s − 0.129·55-s + 0.623·59-s + 1.59·61-s + 0.340·65-s + 1.11·67-s − 0.197·71-s + 1.44·73-s + 0.431·77-s + 1.33·79-s − 0.0773·83-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

\(L(1)\) \(\approx\) \(1.754617271\)
\(L(\frac12)\) \(\approx\) \(1.754617271\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.95T + 7T^{2} \)
11 \( 1 - 0.957T + 11T^{2} \)
13 \( 1 + 2.74T + 13T^{2} \)
17 \( 1 + 5.74T + 17T^{2} \)
19 \( 1 + 6.74T + 19T^{2} \)
29 \( 1 + 5.21T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 - 9.12T + 37T^{2} \)
41 \( 1 + 0.252T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 - 4.78T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 9.12T + 67T^{2} \)
71 \( 1 + 1.66T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + 0.704T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.88469628691378613844031044835, −7.17949091525851784806766355185, −6.55032331893661723181794685974, −5.63939587331360595410296630255, −4.87198410337445255620268498573, −4.30825237768670758338342987431, −3.77338329867495042223101201360, −2.23155966342735809752659588312, −2.08182565979903314592751751187, −0.63375782700526148436771818713, 0.63375782700526148436771818713, 2.08182565979903314592751751187, 2.23155966342735809752659588312, 3.77338329867495042223101201360, 4.30825237768670758338342987431, 4.87198410337445255620268498573, 5.63939587331360595410296630255, 6.55032331893661723181794685974, 7.17949091525851784806766355185, 7.88469628691378613844031044835

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