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Properties

Label 2-4560-76.75-c1-0-56
Degree $2$
Conductor $4560$
Sign $0.898 + 0.438i$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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  + 3-s + 5-s − 2.36i·7-s + 9-s + 0.457i·11-s − 2.36i·13-s + 15-s + 7.27·17-s + (3.91 + 1.90i)19-s − 2.36i·21-s + 3.24i·23-s + 25-s + 27-s + 6.94i·29-s − 0.918·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.894i·7-s + 0.333·9-s + 0.137i·11-s − 0.656i·13-s + 0.258·15-s + 1.76·17-s + (0.898 + 0.438i)19-s − 0.516i·21-s + 0.676i·23-s + 0.200·25-s + 0.192·27-s + 1.28i·29-s − 0.164·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 0.898 + 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.976829843\)
\(L(\frac12)\) \(\approx\) \(2.976829843\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 + (-3.91 - 1.90i)T \)
good7 \( 1 + 2.36iT - 7T^{2} \)
11 \( 1 - 0.457iT - 11T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 - 7.27T + 17T^{2} \)
23 \( 1 - 3.24iT - 23T^{2} \)
29 \( 1 - 6.94iT - 29T^{2} \)
31 \( 1 + 0.918T + 31T^{2} \)
37 \( 1 + 2.78iT - 37T^{2} \)
41 \( 1 + 9.84iT - 41T^{2} \)
43 \( 1 + 2.78iT - 43T^{2} \)
47 \( 1 - 13.5iT - 47T^{2} \)
53 \( 1 - 3.24iT - 53T^{2} \)
59 \( 1 + 6.87T + 59T^{2} \)
61 \( 1 + 3.95T + 61T^{2} \)
67 \( 1 - 2.51T + 67T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + 0.517T + 73T^{2} \)
79 \( 1 + 6.35T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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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

−8.144344836787015840907984561193, −7.38346614628611356153150998682, −7.27633853634545128954267605811, −5.90733603089267826882356743713, −5.45623598434619839591388230924, −4.48472951943628881160392508320, −3.43213815892485423887811106463, −3.12812196362361423609160358934, −1.72901787235794688296472763455, −0.940474889656043028858291791565, 1.06712657597452140813574819594, 2.14309881235108311472015986476, 2.88488958536124095851427459413, 3.66439711839121187226220305010, 4.75266829896564221243812272515, 5.45476500922526873116377101546, 6.16589058423006751962681675501, 6.92914217002059825867860279259, 7.84895064536205901161116596995, 8.330757775721099399960861767986

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