Properties

Label 2-3920-1.1-c1-0-25
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  − 3·3-s + 5-s + 6·9-s + 2·11-s + 6·13-s − 3·15-s − 2·17-s + 9·23-s + 25-s − 9·27-s + 3·29-s + 2·31-s − 6·33-s + 8·37-s − 18·39-s − 5·41-s − 43-s + 6·45-s + 8·47-s + 6·51-s + 4·53-s + 2·55-s − 8·59-s − 7·61-s + 6·65-s + 3·67-s − 27·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s + 1.66·13-s − 0.774·15-s − 0.485·17-s + 1.87·23-s + 1/5·25-s − 1.73·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s + 1.31·37-s − 2.88·39-s − 0.780·41-s − 0.152·43-s + 0.894·45-s + 1.16·47-s + 0.840·51-s + 0.549·53-s + 0.269·55-s − 1.04·59-s − 0.896·61-s + 0.744·65-s + 0.366·67-s − 3.25·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.388511446\)
\(L(\frac12)\) \(\approx\) \(1.388511446\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
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

−8.694241677182228172269756154818, −7.44163518009335683346060783404, −6.68764982840299092997554232374, −6.19980485768653950501230229858, −5.67902275006040205840033693057, −4.78319451656118645556724774548, −4.18265651043873632086354730598, −3.02914700582274231510062881672, −1.49229063377086535263762304039, −0.837009286959784474979743136837, 0.837009286959784474979743136837, 1.49229063377086535263762304039, 3.02914700582274231510062881672, 4.18265651043873632086354730598, 4.78319451656118645556724774548, 5.67902275006040205840033693057, 6.19980485768653950501230229858, 6.68764982840299092997554232374, 7.44163518009335683346060783404, 8.694241677182228172269756154818

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