Properties

Label 2-8280-1.1-c1-0-31
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 − 3.55·7-s + 2.79·11-s + 3.73·13-s + 7.83·17-s + 6.27·19-s + 23-s + 25-s − 2.75·29-s − 2.48·31-s + 3.55·35-s − 1.55·37-s − 5.78·41-s − 4.54·43-s + 6.27·47-s + 5.61·49-s − 8.89·53-s − 2.79·55-s − 3.31·59-s + 11.2·61-s − 3.73·65-s + 5.52·67-s − 6.34·71-s + 15.3·73-s − 9.91·77-s − 9.62·79-s − 5.83·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.34·7-s + 0.842·11-s + 1.03·13-s + 1.89·17-s + 1.44·19-s + 0.208·23-s + 0.200·25-s − 0.512·29-s − 0.447·31-s + 0.600·35-s − 0.255·37-s − 0.904·41-s − 0.693·43-s + 0.915·47-s + 0.801·49-s − 1.22·53-s − 0.376·55-s − 0.431·59-s + 1.43·61-s − 0.462·65-s + 0.674·67-s − 0.752·71-s + 1.80·73-s − 1.13·77-s − 1.08·79-s − 0.640·83-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

\(L(1)\) \(\approx\) \(1.872026300\)
\(L(\frac12)\) \(\approx\) \(1.872026300\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.55T + 7T^{2} \)
11 \( 1 - 2.79T + 11T^{2} \)
13 \( 1 - 3.73T + 13T^{2} \)
17 \( 1 - 7.83T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
29 \( 1 + 2.75T + 29T^{2} \)
31 \( 1 + 2.48T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 + 5.78T + 41T^{2} \)
43 \( 1 + 4.54T + 43T^{2} \)
47 \( 1 - 6.27T + 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + 3.31T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 + 6.34T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 9.62T + 79T^{2} \)
83 \( 1 + 5.83T + 83T^{2} \)
89 \( 1 + 0.390T + 89T^{2} \)
97 \( 1 - 0.961T + 97T^{2} \)
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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

−7.73101191830610626860239319492, −7.08991637496526465922986063239, −6.47741999848811681443634605144, −5.74941702805649743865863114056, −5.19507402007869844321271321894, −3.94977482558191168904860297231, −3.42308137265890646370267167357, −3.08949296168369230736063862162, −1.54412907965128953180658996791, −0.72064640569503056221656053999, 0.72064640569503056221656053999, 1.54412907965128953180658996791, 3.08949296168369230736063862162, 3.42308137265890646370267167357, 3.94977482558191168904860297231, 5.19507402007869844321271321894, 5.74941702805649743865863114056, 6.47741999848811681443634605144, 7.08991637496526465922986063239, 7.73101191830610626860239319492

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