Properties

Label 2-5040-1.1-c1-0-58
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 + 4·11-s − 6·13-s − 2·17-s + 25-s − 6·29-s − 8·31-s + 35-s − 10·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 2·53-s + 4·55-s − 8·59-s − 14·61-s − 6·65-s + 12·67-s − 16·71-s + 2·73-s + 4·77-s + 8·79-s + 8·83-s − 2·85-s − 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s − 0.485·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s − 1.04·59-s − 1.79·61-s − 0.744·65-s + 1.46·67-s − 1.89·71-s + 0.234·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s − 0.216·85-s − 1.05·89-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: $\chi_{5040} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−7.69537665846501814851151036032, −7.18442932220920716960195375619, −6.53503531475229539424605190008, −5.60334584590381835154023547959, −4.99605143965867724115284774422, −4.18930851846167852256621728779, −3.33062511491831582375696335365, −2.19590154838649435040150777594, −1.57587941250931515532871434236, 0, 1.57587941250931515532871434236, 2.19590154838649435040150777594, 3.33062511491831582375696335365, 4.18930851846167852256621728779, 4.99605143965867724115284774422, 5.60334584590381835154023547959, 6.53503531475229539424605190008, 7.18442932220920716960195375619, 7.69537665846501814851151036032

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