Properties

Label 2-8820-1.1-c1-0-8
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 − 4.24·11-s − 5.24·13-s + 4.24·17-s − 7·19-s − 4.24·23-s + 25-s + 10.2·29-s + 7.48·31-s − 5.24·37-s − 4.24·41-s − 5.24·43-s + 6·47-s + 8.48·53-s − 4.24·55-s + 1.75·59-s − 12.4·61-s − 5.24·65-s + 3.24·67-s − 12.7·71-s + 0.757·73-s + 11·79-s + 1.75·83-s + 4.24·85-s + 1.75·89-s − 7·95-s + 16.4·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.27·11-s − 1.45·13-s + 1.02·17-s − 1.60·19-s − 0.884·23-s + 0.200·25-s + 1.90·29-s + 1.34·31-s − 0.861·37-s − 0.662·41-s − 0.799·43-s + 0.875·47-s + 1.16·53-s − 0.572·55-s + 0.228·59-s − 1.59·61-s − 0.650·65-s + 0.396·67-s − 1.51·71-s + 0.0886·73-s + 1.23·79-s + 0.192·83-s + 0.460·85-s + 0.186·89-s − 0.718·95-s + 1.67·97-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: $\chi_{8820} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8820,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.459402772\)
\(L(\frac12)\) \(\approx\) \(1.459402772\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 7.48T + 31T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 8.48T + 53T^{2} \)
59 \( 1 - 1.75T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 3.24T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 0.757T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 - 1.75T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−7.83105676710422415285207893752, −7.08241358759785289969996718264, −6.35075759102156449936861472647, −5.70386470058840429189856553935, −4.85698990207285509432897653804, −4.54348538671556260744623334951, −3.28421640942799455539292980423, −2.55068025912105479953401505325, −1.96612411066196486085001043291, −0.55688013296613931566906696318, 0.55688013296613931566906696318, 1.96612411066196486085001043291, 2.55068025912105479953401505325, 3.28421640942799455539292980423, 4.54348538671556260744623334951, 4.85698990207285509432897653804, 5.70386470058840429189856553935, 6.35075759102156449936861472647, 7.08241358759785289969996718264, 7.83105676710422415285207893752

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