Properties

Label 2-5040-5.4-c1-0-63
Degree $2$
Conductor $5040$
Sign $0.894 + 0.447i$
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 + 2i)5-s + i·7-s + 6·11-s − 2i·13-s − 2i·17-s + 4·19-s − 4i·23-s + (−3 − 4i)25-s + 2·29-s + 2·31-s + (−2 − i)35-s − 10i·37-s − 6·41-s − 2i·43-s − 2i·47-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s + 0.377i·7-s + 1.80·11-s − 0.554i·13-s − 0.485i·17-s + 0.917·19-s − 0.834i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s + 0.359·31-s + (−0.338 − 0.169i)35-s − 1.64i·37-s − 0.937·41-s − 0.304i·43-s − 0.291i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.894 + 0.447i$
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: \((2,\ 5040,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892032022\)
\(L(\frac12)\) \(\approx\) \(1.892032022\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
7 \( 1 - iT \)
good11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−8.155807575656471432744686544768, −7.36095340221376516654939139371, −6.74441008379582629140608651839, −6.20162334697453255935169739523, −5.31528950753101128554555254062, −4.34371805850461078144904530455, −3.58317391838343549071027363000, −2.93602546988935205610830885606, −1.88834890380341325451942590923, −0.59585554502381822404249494613, 1.11282886335662339463885988937, 1.54687819882890179495122354880, 3.15858759785414045170008819792, 3.92611637615677545031048206575, 4.46152639399575343392414033538, 5.27594188110129571461131574342, 6.23531266231085847618114554260, 6.83080839185318661308765045518, 7.61092480550294512553826920740, 8.347878784668809701242172101138

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