Properties

Label 2-5040-5.4-c1-0-50
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 + 2·11-s − 2i·13-s + 2·19-s − 8i·23-s + (−3 + 4i)25-s + 2·29-s + 6·31-s + (−2 + i)35-s + 8i·37-s + 10·41-s − 12i·47-s − 49-s − 2i·53-s + ⋯
L(s)  = 1  + (0.447 + 0.894i)5-s + 0.377i·7-s + 0.603·11-s − 0.554i·13-s + 0.458·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s + 0.371·29-s + 1.07·31-s + (−0.338 + 0.169i)35-s + 1.31i·37-s + 1.56·41-s − 1.75i·47-s − 0.142·49-s − 0.274i·53-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

\(L(1)\) \(\approx\) \(2.342273358\)
\(L(\frac12)\) \(\approx\) \(2.342273358\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.295129513648703418563728464478, −7.56493494561611626219427935582, −6.55009333232478299097879497822, −6.41333172168074011557769800479, −5.44214532972921304361593836805, −4.65504719002921205025060855342, −3.65484075242296012289333356383, −2.83111565149064657528697172348, −2.18139691633490299136085795014, −0.867766105126444131128988026296, 0.881645498449283600742640052979, 1.62483642983660553365881613933, 2.72909928330320160194659919123, 3.89628260787520350511351381059, 4.39505619807262416827813399651, 5.32135659856487585816291865948, 5.93448955554269360501734963314, 6.72032754551153958501931044496, 7.54626143581179443777962769437, 8.146996694777704946128525018144

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