Properties

Label 2-5040-1.1-c1-0-49
Degree $2$
Conductor $5040$
Sign $-1$
Analytic cond. $40.2446$
Root an. cond. $6.34386$
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 − 7-s − 4·13-s + 6·17-s − 2·19-s − 6·23-s + 25-s − 2·31-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 49-s − 6·53-s − 12·59-s − 10·61-s − 4·65-s + 4·67-s − 12·71-s − 4·73-s − 8·79-s − 12·83-s + 6·85-s + 6·89-s + 4·91-s − 2·95-s + 8·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 0.359·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s − 0.468·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.419·91-s − 0.205·95-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.62303324730615600555306548237, −7.42453209666565007953264828448, −6.18823131817684849597691730110, −5.89552237162032170404215264997, −4.94981086641525978002182301109, −4.18496018671806441822001764244, −3.19848947825628311453951559016, −2.43711426518665820924570102939, −1.42136912390124255279326542270, 0, 1.42136912390124255279326542270, 2.43711426518665820924570102939, 3.19848947825628311453951559016, 4.18496018671806441822001764244, 4.94981086641525978002182301109, 5.89552237162032170404215264997, 6.18823131817684849597691730110, 7.42453209666565007953264828448, 7.62303324730615600555306548237

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