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Properties

Label 2-8820-1.1-c1-0-46
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 + 4·11-s − 4·13-s − 2·17-s + 4·23-s + 25-s − 8·31-s + 2·37-s + 2·41-s + 4·43-s − 4·47-s − 4·53-s − 4·55-s − 4·59-s + 8·61-s + 4·65-s − 4·67-s − 12·71-s + 4·73-s + 4·79-s + 8·83-s + 2·85-s − 14·89-s + 12·97-s − 10·101-s − 8·103-s − 12·107-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s − 1.10·13-s − 0.485·17-s + 0.834·23-s + 1/5·25-s − 1.43·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.583·47-s − 0.549·53-s − 0.539·55-s − 0.520·59-s + 1.02·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s + 0.468·73-s + 0.450·79-s + 0.878·83-s + 0.216·85-s − 1.48·89-s + 1.21·97-s − 0.995·101-s − 0.788·103-s − 1.16·107-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: Trivial
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.30672828993234333599589394197, −6.87644128724106102673488561052, −6.11978076235918333081154502407, −5.25505689530664406506123814516, −4.55414188852397690870022194512, −3.91658753603146906810952441844, −3.11479655240398437987449681020, −2.20747071617922617088231186769, −1.21800253363110531001071950669, 0, 1.21800253363110531001071950669, 2.20747071617922617088231186769, 3.11479655240398437987449681020, 3.91658753603146906810952441844, 4.55414188852397690870022194512, 5.25505689530664406506123814516, 6.11978076235918333081154502407, 6.87644128724106102673488561052, 7.30672828993234333599589394197

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