Properties

Label 2-5040-5.4-c1-0-43
Degree $2$
Conductor $5040$
Sign $0.621 - 0.783i$
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  + (1.75 + 1.38i)5-s + i·7-s + 5.14·11-s − 4.64i·13-s + 3.86i·17-s − 0.778·19-s + 5.00i·23-s + (1.14 + 4.86i)25-s + 9.42·29-s − 4.72·31-s + (−1.38 + 1.75i)35-s − 6i·37-s + 1.00·41-s + 7.00i·43-s + 11.4i·47-s + ⋯
L(s)  = 1  + (0.783 + 0.621i)5-s + 0.377i·7-s + 1.55·11-s − 1.28i·13-s + 0.938i·17-s − 0.178·19-s + 1.04i·23-s + (0.228 + 0.973i)25-s + 1.74·29-s − 0.848·31-s + (−0.234 + 0.296i)35-s − 0.986i·37-s + 0.157·41-s + 1.06i·43-s + 1.66i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.621 - 0.783i$
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: \((2,\ 5040,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.607336978\)
\(L(\frac12)\) \(\approx\) \(2.607336978\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 - 1.38i)T \)
7 \( 1 - iT \)
good11 \( 1 - 5.14T + 11T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 - 3.86iT - 17T^{2} \)
19 \( 1 + 0.778T + 19T^{2} \)
23 \( 1 - 5.00iT - 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + 4.72T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 1.00T + 41T^{2} \)
43 \( 1 - 7.00iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 11.7iT - 67T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 - 5.00iT - 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 4.67iT - 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 - 1.58iT - 97T^{2} \)
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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

−8.335932688048614638319578225584, −7.63378752942759123603631807802, −6.72585177051914066449073314119, −6.14705170663295595828534243750, −5.67577181160257812521722894907, −4.70541564934849809133295399980, −3.65234984488424152752519593998, −3.05790242041639887636274229613, −1.99899844331126215284848332938, −1.13581542292539848872261544992, 0.795447834372738572738091558766, 1.66290796688459066011322178459, 2.56419543747605472563613340376, 3.80299912107623642997805773378, 4.47413154981741263961713073683, 5.06038119015582676546599806951, 6.15476259383559822723593834326, 6.67465233480938087366599119878, 7.14885180535969910279049446185, 8.439806621366270092840899551470

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