Properties

Label 2-8280-1.1-c1-0-24
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 + 0.636·7-s − 1.50·11-s − 3.50·13-s + 0.636·17-s − 0.778·19-s − 23-s + 25-s − 4.14·29-s + 2.14·31-s + 0.636·35-s + 1.36·37-s + 1.85·41-s − 7.00·43-s + 3.22·47-s − 6.59·49-s + 0.636·53-s − 1.50·55-s + 11.1·59-s + 13.7·61-s − 3.50·65-s + 13.9·67-s + 8.97·71-s + 15.2·73-s − 0.957·77-s − 12.5·79-s + 9.64·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.240·7-s − 0.453·11-s − 0.972·13-s + 0.154·17-s − 0.178·19-s − 0.208·23-s + 0.200·25-s − 0.769·29-s + 0.384·31-s + 0.107·35-s + 0.224·37-s + 0.290·41-s − 1.06·43-s + 0.469·47-s − 0.942·49-s + 0.0874·53-s − 0.202·55-s + 1.45·59-s + 1.76·61-s − 0.434·65-s + 1.70·67-s + 1.06·71-s + 1.78·73-s − 0.109·77-s − 1.41·79-s + 1.05·83-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

\(L(1)\) \(\approx\) \(1.902259995\)
\(L(\frac12)\) \(\approx\) \(1.902259995\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.636T + 7T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
13 \( 1 + 3.50T + 13T^{2} \)
17 \( 1 - 0.636T + 17T^{2} \)
19 \( 1 + 0.778T + 19T^{2} \)
29 \( 1 + 4.14T + 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 - 1.36T + 37T^{2} \)
41 \( 1 - 1.85T + 41T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 - 0.636T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 - 8.97T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 9.64T + 83T^{2} \)
89 \( 1 + 2.44T + 89T^{2} \)
97 \( 1 + 7.55T + 97T^{2} \)
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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

−7.971476441413637012237946712971, −6.99987345705951829784033869871, −6.58184267633886645665168857692, −5.48251248725833463906869734343, −5.22872147407285261780014487211, −4.32447679290762207484901149499, −3.48223805731530195791643407367, −2.49223560711058444025871478612, −1.94074602795656891406402317046, −0.66122410773765381360007950129, 0.66122410773765381360007950129, 1.94074602795656891406402317046, 2.49223560711058444025871478612, 3.48223805731530195791643407367, 4.32447679290762207484901149499, 5.22872147407285261780014487211, 5.48251248725833463906869734343, 6.58184267633886645665168857692, 6.99987345705951829784033869871, 7.971476441413637012237946712971

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