Properties

Label 2-8820-1.1-c1-0-23
Degree $2$
Conductor $8820$
Sign $1$
Analytic cond. $70.4280$
Root an. cond. $8.39214$
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 − 5.70·11-s + 6.70·13-s + 3.44·17-s − 1.25·19-s + 7.96·23-s + 25-s + 7.18·29-s + 31-s − 8.96·37-s + 10.2·41-s − 4.44·43-s + 1.48·47-s − 6.89·53-s − 5.70·55-s − 0.293·59-s + 0.259·61-s + 6.70·65-s − 6.70·67-s + 11.7·71-s − 0.966·73-s − 11.6·79-s − 14.8·83-s + 3.44·85-s − 12.6·89-s − 1.25·95-s + 13.1·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.72·11-s + 1.86·13-s + 0.836·17-s − 0.288·19-s + 1.66·23-s + 0.200·25-s + 1.33·29-s + 0.179·31-s − 1.47·37-s + 1.59·41-s − 0.678·43-s + 0.216·47-s − 0.947·53-s − 0.769·55-s − 0.0381·59-s + 0.0332·61-s + 0.831·65-s − 0.819·67-s + 1.38·71-s − 0.113·73-s − 1.30·79-s − 1.63·83-s + 0.373·85-s − 1.33·89-s − 0.129·95-s + 1.33·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8820\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.4280\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8820,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398125691\)
\(L(\frac12)\) \(\approx\) \(2.398125691\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 5.70T + 11T^{2} \)
13 \( 1 - 6.70T + 13T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + 1.25T + 19T^{2} \)
23 \( 1 - 7.96T + 23T^{2} \)
29 \( 1 - 7.18T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 8.96T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 - 1.48T + 47T^{2} \)
53 \( 1 + 6.89T + 53T^{2} \)
59 \( 1 + 0.293T + 59T^{2} \)
61 \( 1 - 0.259T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 0.966T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.82608281882477937572049119520, −7.05423311366630226560021147127, −6.32186903228408272189836045815, −5.63876769759902442330980846136, −5.13231969947713118868463027014, −4.31119341901544834550789817450, −3.20079152178332783728306089569, −2.85269382290623237706349708734, −1.66435893573681894709286992916, −0.78359229042255605074944913206, 0.78359229042255605074944913206, 1.66435893573681894709286992916, 2.85269382290623237706349708734, 3.20079152178332783728306089569, 4.31119341901544834550789817450, 5.13231969947713118868463027014, 5.63876769759902442330980846136, 6.32186903228408272189836045815, 7.05423311366630226560021147127, 7.82608281882477937572049119520

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