Properties

Label 2-8820-1.1-c1-0-36
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 1.82·11-s − 6.41·13-s + 3.58·17-s + 7.65·19-s − 3.41·23-s + 25-s + 4.65·29-s − 7.41·31-s − 0.585·37-s + 3.41·41-s + 0.343·43-s + 10.8·47-s + 12.2·53-s + 1.82·55-s + 0.585·59-s − 10.8·61-s + 6.41·65-s + 3.07·67-s + 10.4·71-s − 10.8·73-s − 15.1·79-s − 8·83-s − 3.58·85-s − 16.9·89-s − 7.65·95-s + 9.72·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.551·11-s − 1.77·13-s + 0.869·17-s + 1.75·19-s − 0.711·23-s + 0.200·25-s + 0.864·29-s − 1.33·31-s − 0.0963·37-s + 0.533·41-s + 0.0523·43-s + 1.58·47-s + 1.68·53-s + 0.246·55-s + 0.0762·59-s − 1.38·61-s + 0.795·65-s + 0.375·67-s + 1.24·71-s − 1.26·73-s − 1.70·79-s − 0.878·83-s − 0.388·85-s − 1.79·89-s − 0.785·95-s + 0.987·97-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: \(1\)
Selberg data: \((2,\ 8820,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.82T + 11T^{2} \)
13 \( 1 + 6.41T + 13T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 + 3.41T + 23T^{2} \)
29 \( 1 - 4.65T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 + 0.585T + 37T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 - 0.343T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 0.585T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 3.07T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 9.72T + 97T^{2} \)
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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

−7.39147256907009360363995972721, −7.10379202550673042638583436948, −5.79063171936550874462078254278, −5.39548232140894837777117422221, −4.67304589246096366563352339139, −3.85243241081626781515390275987, −2.98000595675622198442627654925, −2.38412671210003606905580618141, −1.13345424901774665842978120655, 0, 1.13345424901774665842978120655, 2.38412671210003606905580618141, 2.98000595675622198442627654925, 3.85243241081626781515390275987, 4.67304589246096366563352339139, 5.39548232140894837777117422221, 5.79063171936550874462078254278, 7.10379202550673042638583436948, 7.39147256907009360363995972721

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