Properties

Label 2-8280-1.1-c1-0-39
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 − 2.26·7-s − 0.162·11-s + 7.01·13-s + 3.86·17-s + 1.83·19-s + 23-s + 25-s + 7.93·29-s − 5.15·31-s − 2.26·35-s + 6.51·37-s − 1.28·41-s − 5.17·43-s − 4.41·47-s − 1.88·49-s + 0.792·53-s − 0.162·55-s − 4.75·59-s + 13.0·61-s + 7.01·65-s − 1.99·67-s + 11.9·71-s − 11.9·73-s + 0.368·77-s + 9.21·79-s + 7.90·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.854·7-s − 0.0491·11-s + 1.94·13-s + 0.936·17-s + 0.421·19-s + 0.208·23-s + 0.200·25-s + 1.47·29-s − 0.925·31-s − 0.382·35-s + 1.07·37-s − 0.200·41-s − 0.789·43-s − 0.644·47-s − 0.269·49-s + 0.108·53-s − 0.0219·55-s − 0.618·59-s + 1.67·61-s + 0.870·65-s − 0.243·67-s + 1.41·71-s − 1.39·73-s + 0.0419·77-s + 1.03·79-s + 0.867·83-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

\(L(1)\) \(\approx\) \(2.433098195\)
\(L(\frac12)\) \(\approx\) \(2.433098195\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 + 0.162T + 11T^{2} \)
13 \( 1 - 7.01T + 13T^{2} \)
17 \( 1 - 3.86T + 17T^{2} \)
19 \( 1 - 1.83T + 19T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 5.15T + 31T^{2} \)
37 \( 1 - 6.51T + 37T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 + 5.17T + 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 - 0.792T + 53T^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 1.99T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 9.21T + 79T^{2} \)
83 \( 1 - 7.90T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 0.560T + 97T^{2} \)
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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

−7.987597150546807808082413469661, −6.86899219736573873617212727604, −6.46441871027320960374710462849, −5.78898508095145441289812759005, −5.20359496888421860730403541818, −4.12573297826027897393181891864, −3.39754294267740865089801568863, −2.87071965778548710600295564363, −1.61607583415162811502754656433, −0.817989701220327307007684618086, 0.817989701220327307007684618086, 1.61607583415162811502754656433, 2.87071965778548710600295564363, 3.39754294267740865089801568863, 4.12573297826027897393181891864, 5.20359496888421860730403541818, 5.78898508095145441289812759005, 6.46441871027320960374710462849, 6.86899219736573873617212727604, 7.987597150546807808082413469661

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