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Properties

Label 2-360-72.11-c1-0-18
Degree $2$
Conductor $360$
Sign $0.926 - 0.377i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (1.37 + 0.315i)2-s + (−1.57 − 0.720i)3-s + (1.80 + 0.870i)4-s + (0.5 + 0.866i)5-s + (−1.94 − 1.48i)6-s + (−1.68 − 0.970i)7-s + (2.20 + 1.76i)8-s + (1.96 + 2.26i)9-s + (0.415 + 1.35i)10-s + (4.90 + 2.83i)11-s + (−2.20 − 2.66i)12-s + (4.57 − 2.63i)13-s + (−2.01 − 1.86i)14-s + (−0.164 − 1.72i)15-s + (2.48 + 3.13i)16-s + 1.68i·17-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)2-s + (−0.909 − 0.415i)3-s + (0.900 + 0.435i)4-s + (0.223 + 0.387i)5-s + (−0.793 − 0.608i)6-s + (−0.635 − 0.366i)7-s + (0.780 + 0.625i)8-s + (0.654 + 0.756i)9-s + (0.131 + 0.427i)10-s + (1.47 + 0.853i)11-s + (−0.637 − 0.770i)12-s + (1.26 − 0.731i)13-s + (−0.537 − 0.499i)14-s + (−0.0423 − 0.445i)15-s + (0.621 + 0.783i)16-s + 0.409i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.926 - 0.377i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.926 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94131 + 0.380448i\)
\(L(\frac12)\) \(\approx\) \(1.94131 + 0.380448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.315i)T \)
3 \( 1 + (1.57 + 0.720i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (1.68 + 0.970i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.90 - 2.83i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.57 + 2.63i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.68iT - 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
23 \( 1 + (3.68 + 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.72 + 2.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (9.12 - 5.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.70iT - 37T^{2} \)
41 \( 1 + (0.985 - 0.569i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.18 + 3.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.78 + 3.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 + (-4.64 + 2.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.02 + 4.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.52 + 4.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.47T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + (2.53 + 1.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.94 + 2.85i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 + (-1.57 + 2.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.70613095707987875323358839013, −10.74353521090079861389846340458, −10.12203899148258359539639226647, −8.476212680624573606437638904653, −7.16716312922035139077362171859, −6.46677134621760989185433959397, −5.96209173745778055090894467715, −4.52523102971185484277688059501, −3.55190704694475557499819056943, −1.72730452711808031031199807984, 1.43653699769347064105497934400, 3.57657362337085350528653500448, 4.21831951055104719776115941815, 5.82133233053499796283047479319, 5.99246497183270767137648200723, 7.08551707068261966566743409922, 9.006513811880216660852090595638, 9.576490132972802242365248984952, 10.93948848203653673870110638936, 11.42064891433601173652517117366

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