Properties

Label 2-9920-1.1-c1-0-43
Degree $2$
Conductor $9920$
Sign $1$
Analytic cond. $79.2115$
Root an. cond. $8.90008$
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  − 2·3-s + 5-s + 9-s − 2·11-s − 2·15-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 4·27-s + 4·29-s − 31-s + 4·33-s + 8·37-s + 6·41-s − 2·43-s + 45-s − 7·49-s − 4·51-s − 8·53-s − 2·55-s − 8·57-s − 8·59-s − 4·67-s + 8·69-s + 6·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.742·29-s − 0.179·31-s + 0.696·33-s + 1.31·37-s + 0.937·41-s − 0.304·43-s + 0.149·45-s − 49-s − 0.560·51-s − 1.09·53-s − 0.269·55-s − 1.05·57-s − 1.04·59-s − 0.488·67-s + 0.963·69-s + 0.702·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9920\)    =    \(2^{6} \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(79.2115\)
Root analytic conductor: \(8.90008\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.181607303\)
\(L(\frac12)\) \(\approx\) \(1.181607303\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
31 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.64922737077090565350897186292, −6.84037272626182182026356872752, −5.94939111103903493515414346843, −5.89408266018881514174170183139, −4.95638569372654383517660061859, −4.52030469160325774192844738108, −3.33472876047482802668772688457, −2.62625624888861871112114572958, −1.51479246882211507099739763644, −0.56810740418014069389079114466, 0.56810740418014069389079114466, 1.51479246882211507099739763644, 2.62625624888861871112114572958, 3.33472876047482802668772688457, 4.52030469160325774192844738108, 4.95638569372654383517660061859, 5.89408266018881514174170183139, 5.94939111103903493515414346843, 6.84037272626182182026356872752, 7.64922737077090565350897186292

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