Properties

Label 2-5040-5.4-c1-0-57
Degree $2$
Conductor $5040$
Sign $0.994 - 0.100i$
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  + (0.224 + 2.22i)5-s i·7-s + 4.89·11-s − 0.449i·13-s + 2i·17-s + 6.44·19-s − 6.89i·23-s + (−4.89 + i)25-s − 2.89·29-s + 0.898·31-s + (2.22 − 0.224i)35-s − 2i·37-s + 10.8·41-s − 8.89i·43-s + 0.898i·47-s + ⋯
L(s)  = 1  + (0.100 + 0.994i)5-s − 0.377i·7-s + 1.47·11-s − 0.124i·13-s + 0.485i·17-s + 1.47·19-s − 1.43i·23-s + (−0.979 + 0.200i)25-s − 0.538·29-s + 0.161·31-s + (0.376 − 0.0379i)35-s − 0.328i·37-s + 1.70·41-s − 1.35i·43-s + 0.131i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.333592017\)
\(L(\frac12)\) \(\approx\) \(2.333592017\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.224 - 2.22i)T \)
7 \( 1 + iT \)
good11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 0.449iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 6.44T + 19T^{2} \)
23 \( 1 + 6.89iT - 23T^{2} \)
29 \( 1 + 2.89T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 8.89iT - 43T^{2} \)
47 \( 1 - 0.898iT - 47T^{2} \)
53 \( 1 - 1.10iT - 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 6.89iT - 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−8.171298761262340544728326923704, −7.34210703640813214038651098034, −6.88244484459048770974445788357, −6.17617089251364066536423188788, −5.51479384292155287184623484691, −4.31283406497331057958055738986, −3.76187453571044554254606855787, −2.94593466857595596927338798219, −1.94940245949323781430965355352, −0.814140829984474392107994129268, 0.976647197592900227250172239248, 1.61381759945903315702740847014, 2.88607829786779263089333182912, 3.83580838139339385885178922686, 4.49044459110839190857760679810, 5.47099251331809770855341054474, 5.80532248480771422934290188178, 6.86209770778167519149121969280, 7.51451823150925472458967499557, 8.295674125336074724000199208397

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