Properties

Label 2-5040-105.104-c1-0-82
Degree $2$
Conductor $5040$
Sign $0.534 + 0.844i$
Analytic cond. $40.2446$
Root an. cond. $6.34386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.95 + 1.08i)5-s + (2.37 − 1.16i)7-s − 3.74i·11-s − 0.841·13-s − 3.36i·17-s − 4.55i·19-s + 7.64·23-s + (2.64 + 4.24i)25-s + 1.41i·29-s + 0.979i·31-s + (5.90 + 0.302i)35-s + 2.32i·37-s − 10.3·41-s − 10.8i·43-s + 7.91i·47-s + ⋯
L(s)  = 1  + (0.874 + 0.485i)5-s + (0.898 − 0.439i)7-s − 1.12i·11-s − 0.233·13-s − 0.814i·17-s − 1.04i·19-s + 1.59·23-s + (0.529 + 0.848i)25-s + 0.262i·29-s + 0.175i·31-s + (0.998 + 0.0511i)35-s + 0.382i·37-s − 1.61·41-s − 1.64i·43-s + 1.15i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.534 + 0.844i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 0.534 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.570309645\)
\(L(\frac12)\) \(\approx\) \(2.570309645\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 1.08i)T \)
7 \( 1 + (-2.37 + 1.16i)T \)
good11 \( 1 + 3.74iT - 11T^{2} \)
13 \( 1 + 0.841T + 13T^{2} \)
17 \( 1 + 3.36iT - 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 - 7.64T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 0.979iT - 31T^{2} \)
37 \( 1 - 2.32iT - 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 - 7.91iT - 47T^{2} \)
53 \( 1 - 4.35T + 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 + 3.74iT - 71T^{2} \)
73 \( 1 + 8.66T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + 3.14iT - 83T^{2} \)
89 \( 1 + 3.91T + 89T^{2} \)
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

−8.177575914629031247831610706069, −7.10424858836033194922654455518, −6.94965694204166123752596851295, −5.88224937194621295880180662796, −5.15154840142298151261748344424, −4.67409850321802730196223403937, −3.34675161672127765874409365845, −2.79609560091490400178632382479, −1.71521315067225872031202252183, −0.69793045350836045068934048992, 1.32062050111537520869737443308, 1.88586946859721525834265532351, 2.78598769707898161288629534564, 4.06385120532056658945236785862, 4.81126407447401152324438465202, 5.36213024117373477799216251570, 6.06528409170182575263402041868, 6.94111470045227842058872811593, 7.65680502816620203751291084573, 8.540070071385991476928433527901

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