Properties

Label 2-5040-1.1-c1-0-13
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 − 1.56·11-s + 0.438·13-s + 0.438·17-s + 7.12·19-s + 3.12·23-s + 25-s − 6.68·29-s − 35-s + 6·37-s − 5.12·41-s − 0.876·43-s − 8.68·47-s + 49-s + 5.12·53-s + 1.56·55-s − 4·59-s + 15.3·61-s − 0.438·65-s − 10.2·67-s + 8·71-s − 12.2·73-s − 1.56·77-s + 2.43·79-s + 4·83-s − 0.438·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.470·11-s + 0.121·13-s + 0.106·17-s + 1.63·19-s + 0.651·23-s + 0.200·25-s − 1.24·29-s − 0.169·35-s + 0.986·37-s − 0.800·41-s − 0.133·43-s − 1.26·47-s + 0.142·49-s + 0.703·53-s + 0.210·55-s − 0.520·59-s + 1.96·61-s − 0.0543·65-s − 1.25·67-s + 0.949·71-s − 1.43·73-s − 0.177·77-s + 0.274·79-s + 0.439·83-s − 0.0475·85-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

\(L(1)\) \(\approx\) \(1.801145127\)
\(L(\frac12)\) \(\approx\) \(1.801145127\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
17 \( 1 - 0.438T + 17T^{2} \)
19 \( 1 - 7.12T + 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + 0.876T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 5.12T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 2.43T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 - 5.80T + 97T^{2} \)
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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

−8.142714943473368684993814615183, −7.51371198676106616266290266055, −7.02971819149202655701404643524, −5.96324927266163111427330138227, −5.26176791022635776098895890764, −4.65007125380759800348773691451, −3.61858962062271477339498685643, −3.00305092159029934898427042191, −1.85410117085905040543520141482, −0.74422336636659750129507060410, 0.74422336636659750129507060410, 1.85410117085905040543520141482, 3.00305092159029934898427042191, 3.61858962062271477339498685643, 4.65007125380759800348773691451, 5.26176791022635776098895890764, 5.96324927266163111427330138227, 7.02971819149202655701404643524, 7.51371198676106616266290266055, 8.142714943473368684993814615183

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