Properties

Label 2-5780-17.16-c1-0-76
Degree $2$
Conductor $5780$
Sign $-0.970 + 0.242i$
Analytic cond. $46.1535$
Root an. cond. $6.79363$
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  − 2i·3-s i·5-s − 2i·7-s − 9-s + 2·13-s − 2·15-s + 4·19-s − 4·21-s − 6i·23-s − 25-s − 4i·27-s + 6i·29-s − 4i·31-s − 2·35-s + 2i·37-s + ⋯
L(s)  = 1  − 1.15i·3-s − 0.447i·5-s − 0.755i·7-s − 0.333·9-s + 0.554·13-s − 0.516·15-s + 0.917·19-s − 0.872·21-s − 1.25i·23-s − 0.200·25-s − 0.769i·27-s + 1.11i·29-s − 0.718i·31-s − 0.338·35-s + 0.328i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5780\)    =    \(2^{2} \cdot 5 \cdot 17^{2}\)
Sign: $-0.970 + 0.242i$
Analytic conductor: \(46.1535\)
Root analytic conductor: \(6.79363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5780} (5201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5780,\ (\ :1/2),\ -0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.937505363\)
\(L(\frac12)\) \(\approx\) \(1.937505363\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
17 \( 1 \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
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

−7.54550727696115930804371067082, −7.27855198407082602559688554469, −6.42964571963659536158206486889, −5.85326470100329636488713623089, −4.85827161112709190911421409391, −4.13598502224233865669736387375, −3.21686565216667850936633096004, −2.15625909029528111671861946318, −1.24702266708740534038997214756, −0.55378584897632678778180131710, 1.32115725402431437071387720035, 2.54698066709630898551387348849, 3.32920445921698485353125905042, 3.97450644175189896761544431501, 4.79908745620068011328590817665, 5.59115023153883955987047102669, 6.03874362819015385246231840984, 7.09424566657196818893143442697, 7.73276061797015930620379998943, 8.616773272986044176881022167148

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