Properties

Label 2-8280-1.1-c1-0-44
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 + 2.71·7-s + 2.76·11-s − 1.01·13-s − 4.50·17-s + 4.76·19-s + 23-s + 25-s + 8.08·29-s − 5.49·31-s + 2.71·35-s − 10.9·37-s − 2.43·41-s + 5.77·43-s + 11.0·47-s + 0.356·49-s + 8.15·53-s + 2.76·55-s − 1.70·59-s + 3.12·61-s − 1.01·65-s + 5.56·67-s − 2.07·71-s + 9.63·73-s + 7.49·77-s − 3.63·79-s + 11.7·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.02·7-s + 0.833·11-s − 0.280·13-s − 1.09·17-s + 1.09·19-s + 0.208·23-s + 0.200·25-s + 1.50·29-s − 0.986·31-s + 0.458·35-s − 1.80·37-s − 0.380·41-s + 0.880·43-s + 1.61·47-s + 0.0509·49-s + 1.11·53-s + 0.372·55-s − 0.221·59-s + 0.400·61-s − 0.125·65-s + 0.679·67-s − 0.246·71-s + 1.12·73-s + 0.854·77-s − 0.409·79-s + 1.29·83-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

\(L(1)\) \(\approx\) \(2.844169869\)
\(L(\frac12)\) \(\approx\) \(2.844169869\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
13 \( 1 + 1.01T + 13T^{2} \)
17 \( 1 + 4.50T + 17T^{2} \)
19 \( 1 - 4.76T + 19T^{2} \)
29 \( 1 - 8.08T + 29T^{2} \)
31 \( 1 + 5.49T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 2.43T + 41T^{2} \)
43 \( 1 - 5.77T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 8.15T + 53T^{2} \)
59 \( 1 + 1.70T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 5.56T + 67T^{2} \)
71 \( 1 + 2.07T + 71T^{2} \)
73 \( 1 - 9.63T + 73T^{2} \)
79 \( 1 + 3.63T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 4.13T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−7.77613123578000845192872578522, −7.03464312280223326771081121724, −6.58358677270305216707555746942, −5.57381143784980566143549943265, −5.08055642214788381092113862290, −4.34634240116177246886416110852, −3.56120644817169924573871883052, −2.50225248407168403152636984477, −1.76520432537489537850685469556, −0.871246311518717853347318950590, 0.871246311518717853347318950590, 1.76520432537489537850685469556, 2.50225248407168403152636984477, 3.56120644817169924573871883052, 4.34634240116177246886416110852, 5.08055642214788381092113862290, 5.57381143784980566143549943265, 6.58358677270305216707555746942, 7.03464312280223326771081121724, 7.77613123578000845192872578522

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