Properties

Label 2-8280-1.1-c1-0-2
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 − 5.02·7-s + 3.71·11-s − 6.49·13-s − 5.37·17-s − 8.39·19-s + 23-s + 25-s − 3.31·29-s + 7.18·31-s + 5.02·35-s − 3.02·37-s − 0.782·41-s − 0.0950·43-s − 8.39·47-s + 18.2·49-s − 6.82·53-s − 3.71·55-s − 11.2·59-s − 14.6·61-s + 6.49·65-s − 9.71·67-s − 8.73·71-s + 3.64·73-s − 18.6·77-s + 8.59·79-s + 7.37·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.89·7-s + 1.11·11-s − 1.80·13-s − 1.30·17-s − 1.92·19-s + 0.208·23-s + 0.200·25-s − 0.615·29-s + 1.28·31-s + 0.848·35-s − 0.496·37-s − 0.122·41-s − 0.0144·43-s − 1.22·47-s + 2.60·49-s − 0.937·53-s − 0.500·55-s − 1.46·59-s − 1.86·61-s + 0.805·65-s − 1.18·67-s − 1.03·71-s + 0.426·73-s − 2.12·77-s + 0.966·79-s + 0.809·83-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

\(L(1)\) \(\approx\) \(0.2270308955\)
\(L(\frac12)\) \(\approx\) \(0.2270308955\)
\(L(\frac{3}{2})\) 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 \( 1 + 5.02T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 + 8.39T + 19T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 - 7.18T + 31T^{2} \)
37 \( 1 + 3.02T + 37T^{2} \)
41 \( 1 + 0.782T + 41T^{2} \)
43 \( 1 + 0.0950T + 43T^{2} \)
47 \( 1 + 8.39T + 47T^{2} \)
53 \( 1 + 6.82T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 9.71T + 67T^{2} \)
71 \( 1 + 8.73T + 71T^{2} \)
73 \( 1 - 3.64T + 73T^{2} \)
79 \( 1 - 8.59T + 79T^{2} \)
83 \( 1 - 7.37T + 83T^{2} \)
89 \( 1 - 6.29T + 89T^{2} \)
97 \( 1 + 5.32T + 97T^{2} \)
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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

−7.67068607014437049866931487630, −6.92431566898028723280880885140, −6.47123630617906074515534728455, −6.11048203850800311622518749798, −4.68852410266224941595303542873, −4.38429776250523491488629837582, −3.43201298988962921435146036445, −2.76802605723353127996773736381, −1.90718926491961647063569073258, −0.21696996601724194680165685759, 0.21696996601724194680165685759, 1.90718926491961647063569073258, 2.76802605723353127996773736381, 3.43201298988962921435146036445, 4.38429776250523491488629837582, 4.68852410266224941595303542873, 6.11048203850800311622518749798, 6.47123630617906074515534728455, 6.92431566898028723280880885140, 7.67068607014437049866931487630

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