Properties

Label 2-8820-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s + 5·13-s + 17-s + 6·19-s + 4·23-s + 25-s − 3·29-s − 2·31-s + 8·37-s − 10·41-s − 2·43-s − 7·47-s + 2·53-s − 55-s + 14·59-s + 8·61-s − 5·65-s + 14·67-s + 10·73-s − 11·79-s − 4·83-s − 85-s + 4·89-s − 6·95-s + 3·97-s − 10·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s + 1.38·13-s + 0.242·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 1.31·37-s − 1.56·41-s − 0.304·43-s − 1.02·47-s + 0.274·53-s − 0.134·55-s + 1.82·59-s + 1.02·61-s − 0.620·65-s + 1.71·67-s + 1.17·73-s − 1.23·79-s − 0.439·83-s − 0.108·85-s + 0.423·89-s − 0.615·95-s + 0.304·97-s − 0.995·101-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: \(0\)
Selberg data: \((2,\ 8820,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.256456748\)
\(L(\frac12)\) \(\approx\) \(2.256456748\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−7.897896371463336345266991990935, −6.92744854755746429975089901874, −6.59474597232800625069678245466, −5.51589090779766153294605834687, −5.17174879937803300953760703666, −4.00544317776361456006542139262, −3.58722367743226563989338538796, −2.79857499378462339469274671475, −1.55783645740042646619385285491, −0.78870727948012110458706875692, 0.78870727948012110458706875692, 1.55783645740042646619385285491, 2.79857499378462339469274671475, 3.58722367743226563989338538796, 4.00544317776361456006542139262, 5.17174879937803300953760703666, 5.51589090779766153294605834687, 6.59474597232800625069678245466, 6.92744854755746429975089901874, 7.897896371463336345266991990935

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