Properties

Label 2-8820-1.1-c1-0-51
Degree $2$
Conductor $8820$
Sign $-1$
Analytic cond. $70.4280$
Root an. cond. $8.39214$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·11-s + 13-s + 4·17-s − 19-s − 4·23-s + 25-s − 5·31-s − 5·37-s − 2·41-s − 9·43-s + 2·47-s − 12·53-s − 2·55-s + 8·59-s − 14·61-s − 65-s + 9·67-s − 2·71-s + 73-s − 3·79-s + 18·83-s − 4·85-s + 4·89-s + 95-s + 10·97-s − 6·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.603·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.898·31-s − 0.821·37-s − 0.312·41-s − 1.37·43-s + 0.291·47-s − 1.64·53-s − 0.269·55-s + 1.04·59-s − 1.79·61-s − 0.124·65-s + 1.09·67-s − 0.237·71-s + 0.117·73-s − 0.337·79-s + 1.97·83-s − 0.433·85-s + 0.423·89-s + 0.102·95-s + 1.01·97-s − 0.597·101-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: yes
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.51441444199240092505867603473, −6.68723194138330302055960411300, −6.13107958140764687931896593345, −5.30600412310988084273065070739, −4.61699613247306940342757957150, −3.65457817257208641149655015126, −3.36711943082168207493243156146, −2.10032184305076605234204631432, −1.26446692098214953098649762370, 0, 1.26446692098214953098649762370, 2.10032184305076605234204631432, 3.36711943082168207493243156146, 3.65457817257208641149655015126, 4.61699613247306940342757957150, 5.30600412310988084273065070739, 6.13107958140764687931896593345, 6.68723194138330302055960411300, 7.51441444199240092505867603473

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