Properties

Label 2-5040-1.1-c1-0-24
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 + 2.56·11-s + 4.56·13-s + 4.56·17-s − 1.12·19-s − 5.12·23-s + 25-s + 5.68·29-s − 35-s + 6·37-s + 3.12·41-s − 9.12·43-s + 3.68·47-s + 49-s − 3.12·53-s − 2.56·55-s − 4·59-s − 9.36·61-s − 4.56·65-s + 6.24·67-s + 8·71-s + 4.24·73-s + 2.56·77-s + 6.56·79-s + 4·83-s − 4.56·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.772·11-s + 1.26·13-s + 1.10·17-s − 0.257·19-s − 1.06·23-s + 0.200·25-s + 1.05·29-s − 0.169·35-s + 0.986·37-s + 0.487·41-s − 1.39·43-s + 0.537·47-s + 0.142·49-s − 0.428·53-s − 0.345·55-s − 0.520·59-s − 1.19·61-s − 0.565·65-s + 0.763·67-s + 0.949·71-s + 0.496·73-s + 0.291·77-s + 0.738·79-s + 0.439·83-s − 0.494·85-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: $\chi_{5040} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207465479\)
\(L(\frac12)\) \(\approx\) \(2.207465479\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 4.56T + 17T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 - 5.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 + 3.12T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 9.36T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 - 6.56T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 7.12T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.122446512639030530693895560621, −7.76860583892026575047000319187, −6.65645291240195735906747375761, −6.18832360619498556644098522520, −5.35335882065032152235231234975, −4.37791204752379988825121286000, −3.80471134928957207647720778826, −3.00601330755127367252021836079, −1.73494510177470132307942603571, −0.867016930643069539805314838033, 0.867016930643069539805314838033, 1.73494510177470132307942603571, 3.00601330755127367252021836079, 3.80471134928957207647720778826, 4.37791204752379988825121286000, 5.35335882065032152235231234975, 6.18832360619498556644098522520, 6.65645291240195735906747375761, 7.76860583892026575047000319187, 8.122446512639030530693895560621

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