Properties

Label 2-8280-1.1-c1-0-28
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 + 2.38·7-s − 5.33·11-s − 4.53·13-s − 1.81·17-s + 7.00·19-s + 23-s + 25-s + 0.118·29-s − 0.884·31-s + 2.38·35-s + 7.51·37-s + 1.45·41-s − 10.4·47-s − 1.32·49-s + 9.42·53-s − 5.33·55-s − 7.79·59-s − 2.80·61-s − 4.53·65-s − 3.11·67-s + 13.5·71-s + 12.4·73-s − 12.7·77-s + 6.80·79-s + 13.5·83-s − 1.81·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.900·7-s − 1.60·11-s − 1.25·13-s − 0.440·17-s + 1.60·19-s + 0.208·23-s + 0.200·25-s + 0.0219·29-s − 0.158·31-s + 0.402·35-s + 1.23·37-s + 0.226·41-s − 1.52·47-s − 0.189·49-s + 1.29·53-s − 0.719·55-s − 1.01·59-s − 0.359·61-s − 0.562·65-s − 0.380·67-s + 1.61·71-s + 1.45·73-s − 1.44·77-s + 0.765·79-s + 1.48·83-s − 0.196·85-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

\(L(1)\) \(\approx\) \(2.020589064\)
\(L(\frac12)\) \(\approx\) \(2.020589064\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 + 5.33T + 11T^{2} \)
13 \( 1 + 4.53T + 13T^{2} \)
17 \( 1 + 1.81T + 17T^{2} \)
19 \( 1 - 7.00T + 19T^{2} \)
29 \( 1 - 0.118T + 29T^{2} \)
31 \( 1 + 0.884T + 31T^{2} \)
37 \( 1 - 7.51T + 37T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 9.42T + 53T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 + 2.80T + 61T^{2} \)
67 \( 1 + 3.11T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 6.80T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + 2.89T + 89T^{2} \)
97 \( 1 + 1.97T + 97T^{2} \)
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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

−7.75439918450227469210137170157, −7.35566462744005159835792049977, −6.42986413431045574438234186874, −5.47480578126739390116645691687, −5.05060444451028939113916043167, −4.60844251710145976019409731613, −3.31359731956540054699556897469, −2.55790502968397206883568069156, −1.92278472834604443669717788447, −0.68406165400172018254651081345, 0.68406165400172018254651081345, 1.92278472834604443669717788447, 2.55790502968397206883568069156, 3.31359731956540054699556897469, 4.60844251710145976019409731613, 5.05060444451028939113916043167, 5.47480578126739390116645691687, 6.42986413431045574438234186874, 7.35566462744005159835792049977, 7.75439918450227469210137170157

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