Properties

Label 2-5040-1.1-c1-0-32
Degree $2$
Conductor $5040$
Sign $1$
Analytic cond. $40.2446$
Root an. cond. $6.34386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 6.37·11-s + 4.37·13-s + 0.372·17-s + 4.74·19-s − 4.74·23-s + 25-s + 4.37·29-s + 8·31-s + 35-s − 2·37-s − 6.74·41-s + 8.74·43-s − 7.11·47-s + 49-s − 10.7·53-s + 6.37·55-s + 8·59-s − 2.74·61-s + 4.37·65-s + 4·67-s + 8·71-s − 6·73-s + 6.37·77-s − 15.1·79-s − 9.48·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.92·11-s + 1.21·13-s + 0.0902·17-s + 1.08·19-s − 0.989·23-s + 0.200·25-s + 0.811·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s − 1.05·41-s + 1.33·43-s − 1.03·47-s + 0.142·49-s − 1.47·53-s + 0.859·55-s + 1.04·59-s − 0.351·61-s + 0.542·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.726·77-s − 1.70·79-s − 1.04·83-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.000001631\)
\(L(\frac12)\) \(\approx\) \(3.000001631\)
\(L(\frac{3}{2})\) 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 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 - 4.37T + 13T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 + 4.74T + 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.74T + 41T^{2} \)
43 \( 1 - 8.74T + 43T^{2} \)
47 \( 1 + 7.11T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 2.74T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + 9.86T + 97T^{2} \)
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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

−8.467059996879404525947152227668, −7.51389081844806909571369654602, −6.53898130747148901259195148314, −6.27445690166024617438760896820, −5.40039871596592190389876787406, −4.43328872388879689377525781247, −3.79290196525935341569016368854, −2.93368673587216142483582339641, −1.61640183045409330966667569580, −1.09891444278881662790486588744, 1.09891444278881662790486588744, 1.61640183045409330966667569580, 2.93368673587216142483582339641, 3.79290196525935341569016368854, 4.43328872388879689377525781247, 5.40039871596592190389876787406, 6.27445690166024617438760896820, 6.53898130747148901259195148314, 7.51389081844806909571369654602, 8.467059996879404525947152227668

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