Properties

Label 2-8280-69.68-c1-0-25
Degree $2$
Conductor $8280$
Sign $0.934 - 0.356i$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 0.488i·7-s − 0.477·11-s − 4.91·13-s + 0.0686·17-s − 2.26i·19-s + (−4.64 − 1.19i)23-s + 25-s + 2.85i·29-s − 3.06·31-s + 0.488i·35-s + 5.16i·37-s − 6.04i·41-s + 2.17i·43-s + 0.440i·47-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.184i·7-s − 0.144·11-s − 1.36·13-s + 0.0166·17-s − 0.520i·19-s + (−0.968 − 0.248i)23-s + 0.200·25-s + 0.530i·29-s − 0.551·31-s + 0.0825i·35-s + 0.849i·37-s − 0.944i·41-s + 0.331i·43-s + 0.0642i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.111174255\)
\(L(\frac12)\) \(\approx\) \(1.111174255\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.64 + 1.19i)T \)
good7 \( 1 + 0.488iT - 7T^{2} \)
11 \( 1 + 0.477T + 11T^{2} \)
13 \( 1 + 4.91T + 13T^{2} \)
17 \( 1 - 0.0686T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
29 \( 1 - 2.85iT - 29T^{2} \)
31 \( 1 + 3.06T + 31T^{2} \)
37 \( 1 - 5.16iT - 37T^{2} \)
41 \( 1 + 6.04iT - 41T^{2} \)
43 \( 1 - 2.17iT - 43T^{2} \)
47 \( 1 - 0.440iT - 47T^{2} \)
53 \( 1 + 5.95T + 53T^{2} \)
59 \( 1 - 2.25iT - 59T^{2} \)
61 \( 1 + 3.28iT - 61T^{2} \)
67 \( 1 + 6.36iT - 67T^{2} \)
71 \( 1 - 8.73iT - 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 9.13iT - 79T^{2} \)
83 \( 1 - 3.27T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 17.5iT - 97T^{2} \)
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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

−7.84213589658825809429297445297, −7.14691034470990092680803166353, −6.67050745671930979282086819159, −5.68458788546723432827724469926, −4.98998697754038830301065634687, −4.37334997246589312610541138604, −3.56197154046437058079367778521, −2.70056500139585107487385955565, −1.91424636576097719461833841838, −0.58293642461789969053709212030, 0.42664289377587884559209230090, 1.84098425833737984608615112065, 2.55301000722412959860850860962, 3.50573289577107633972391006673, 4.21801118232791051453604720193, 4.97336893885644060373894331512, 5.65255389581893670016683547729, 6.39218181565758339005761854990, 7.23179285748256675716540814527, 7.76765905967278772338806178812

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