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Properties

Label 2-360-120.59-c3-0-32
Degree $2$
Conductor $360$
Sign $0.684 + 0.729i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.80 + 0.385i)2-s + (7.70 − 2.16i)4-s + (6.42 − 9.15i)5-s − 30.0·7-s + (−20.7 + 9.02i)8-s + (−14.4 + 28.1i)10-s + 47.7i·11-s + 12.3·13-s + (84.0 − 11.5i)14-s + (54.6 − 33.2i)16-s + 130.·17-s − 19.2·19-s + (29.6 − 84.3i)20-s + (−18.4 − 133. i)22-s + 58.7i·23-s + ⋯
L(s)  = 1  + (−0.990 + 0.136i)2-s + (0.962 − 0.270i)4-s + (0.574 − 0.818i)5-s − 1.62·7-s + (−0.916 + 0.398i)8-s + (−0.457 + 0.889i)10-s + 1.30i·11-s + 0.262·13-s + (1.60 − 0.220i)14-s + (0.853 − 0.520i)16-s + 1.86·17-s − 0.232·19-s + (0.331 − 0.943i)20-s + (−0.178 − 1.29i)22-s + 0.532i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.684 + 0.729i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ 0.684 + 0.729i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.013292928\)
\(L(\frac12)\) \(\approx\) \(1.013292928\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.80 - 0.385i)T \)
3 \( 1 \)
5 \( 1 + (-6.42 + 9.15i)T \)
good7 \( 1 + 30.0T + 343T^{2} \)
11 \( 1 - 47.7iT - 1.33e3T^{2} \)
13 \( 1 - 12.3T + 2.19e3T^{2} \)
17 \( 1 - 130.T + 4.91e3T^{2} \)
19 \( 1 + 19.2T + 6.85e3T^{2} \)
23 \( 1 - 58.7iT - 1.21e4T^{2} \)
29 \( 1 - 176.T + 2.43e4T^{2} \)
31 \( 1 + 171. iT - 2.97e4T^{2} \)
37 \( 1 + 91.9T + 5.06e4T^{2} \)
41 \( 1 + 251. iT - 6.89e4T^{2} \)
43 \( 1 + 48.2iT - 7.95e4T^{2} \)
47 \( 1 + 501. iT - 1.03e5T^{2} \)
53 \( 1 + 222. iT - 1.48e5T^{2} \)
59 \( 1 + 348. iT - 2.05e5T^{2} \)
61 \( 1 + 767. iT - 2.26e5T^{2} \)
67 \( 1 - 176. iT - 3.00e5T^{2} \)
71 \( 1 + 44.0T + 3.57e5T^{2} \)
73 \( 1 - 793. iT - 3.89e5T^{2} \)
79 \( 1 + 930. iT - 4.93e5T^{2} \)
83 \( 1 - 957.T + 5.71e5T^{2} \)
89 \( 1 - 1.25e3iT - 7.04e5T^{2} \)
97 \( 1 - 94.5iT - 9.12e5T^{2} \)
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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

−10.32023900907704440196924171282, −9.843064778280334133988660180441, −9.317696218241163127437174999858, −8.207908087516510224694614050299, −7.13660123544870494520306585456, −6.23602991753196348202135311229, −5.29455214495762549792484928948, −3.49690856017342191891927400262, −2.05493139873883602150454655966, −0.62005513834771747687141056236, 0.981464366237197734483667625436, 2.93774119280739965394695501465, 3.29242928462948340520437034905, 5.92455452723836190800791137761, 6.31171076333462737922273976387, 7.35359540340067774696640365331, 8.508306940181178154923197423677, 9.446647654934481590568986871540, 10.22302012518698359357199748964, 10.70513999778940232798616998801

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