Properties

Label 2-8280-1.1-c1-0-63
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 + 4.12·7-s + 4.64·11-s + 2.64·13-s + 4.12·17-s − 1.60·19-s − 23-s + 25-s − 1.48·29-s − 0.515·31-s + 4.12·35-s − 2.12·37-s + 4.51·41-s + 5.28·43-s + 2.39·47-s + 10.0·49-s + 4.12·53-s + 4.64·55-s − 3.79·59-s + 2.32·61-s + 2.64·65-s − 0.185·67-s + 14.9·71-s − 10.1·73-s + 19.1·77-s − 1.93·79-s + 0.844·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.55·7-s + 1.39·11-s + 0.732·13-s + 1.00·17-s − 0.369·19-s − 0.208·23-s + 0.200·25-s − 0.275·29-s − 0.0925·31-s + 0.697·35-s − 0.349·37-s + 0.705·41-s + 0.805·43-s + 0.348·47-s + 1.43·49-s + 0.566·53-s + 0.625·55-s − 0.494·59-s + 0.298·61-s + 0.327·65-s − 0.0226·67-s + 1.77·71-s − 1.19·73-s + 2.18·77-s − 0.218·79-s + 0.0927·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.586938278\)
\(L(\frac12)\) \(\approx\) \(3.586938278\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.12T + 7T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
29 \( 1 + 1.48T + 29T^{2} \)
31 \( 1 + 0.515T + 31T^{2} \)
37 \( 1 + 2.12T + 37T^{2} \)
41 \( 1 - 4.51T + 41T^{2} \)
43 \( 1 - 5.28T + 43T^{2} \)
47 \( 1 - 2.39T + 47T^{2} \)
53 \( 1 - 4.12T + 53T^{2} \)
59 \( 1 + 3.79T + 59T^{2} \)
61 \( 1 - 2.32T + 61T^{2} \)
67 \( 1 + 0.185T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 1.93T + 79T^{2} \)
83 \( 1 - 0.844T + 83T^{2} \)
89 \( 1 + 0.780T + 89T^{2} \)
97 \( 1 + 9.21T + 97T^{2} \)
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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

−7.86045931477227275667962074965, −7.17269483475898249163540497564, −6.34699984018479211499379168701, −5.72227842405176980325646295639, −5.07906816794478920043471446268, −4.18636207563779113946576057425, −3.72196090887798861878111956788, −2.50158112456511000198698090937, −1.56388609706987020206002817225, −1.08771509451387408035823179124, 1.08771509451387408035823179124, 1.56388609706987020206002817225, 2.50158112456511000198698090937, 3.72196090887798861878111956788, 4.18636207563779113946576057425, 5.07906816794478920043471446268, 5.72227842405176980325646295639, 6.34699984018479211499379168701, 7.17269483475898249163540497564, 7.86045931477227275667962074965

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