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Properties

Label 2-360-120.59-c3-0-31
Degree $2$
Conductor $360$
Sign $0.983 - 0.178i$
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  + (0.778 + 2.71i)2-s + (−6.78 + 4.23i)4-s + (−11.1 − 0.469i)5-s − 17.5·7-s + (−16.8 − 15.1i)8-s + (−7.42 − 30.7i)10-s + 1.21i·11-s − 2.29·13-s + (−13.6 − 47.7i)14-s + (28.1 − 57.4i)16-s + 38.9·17-s + 101.·19-s + (77.7 − 44.1i)20-s + (−3.29 + 0.943i)22-s + 79.2i·23-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (−0.848 + 0.529i)4-s + (−0.999 − 0.0419i)5-s − 0.948·7-s + (−0.742 − 0.669i)8-s + (−0.234 − 0.972i)10-s + 0.0331i·11-s − 0.0490·13-s + (−0.261 − 0.911i)14-s + (0.439 − 0.898i)16-s + 0.555·17-s + 1.22·19-s + (0.869 − 0.493i)20-s + (−0.0319 + 0.00914i)22-s + 0.718i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.983 - 0.178i$
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: \((2,\ 360,\ (\ :3/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9966067227\)
\(L(\frac12)\) \(\approx\) \(0.9966067227\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.778 - 2.71i)T \)
3 \( 1 \)
5 \( 1 + (11.1 + 0.469i)T \)
good7 \( 1 + 17.5T + 343T^{2} \)
11 \( 1 - 1.21iT - 1.33e3T^{2} \)
13 \( 1 + 2.29T + 2.19e3T^{2} \)
17 \( 1 - 38.9T + 4.91e3T^{2} \)
19 \( 1 - 101.T + 6.85e3T^{2} \)
23 \( 1 - 79.2iT - 1.21e4T^{2} \)
29 \( 1 + 188.T + 2.43e4T^{2} \)
31 \( 1 + 195. iT - 2.97e4T^{2} \)
37 \( 1 - 377.T + 5.06e4T^{2} \)
41 \( 1 + 205. iT - 6.89e4T^{2} \)
43 \( 1 + 396. iT - 7.95e4T^{2} \)
47 \( 1 - 429. iT - 1.03e5T^{2} \)
53 \( 1 - 65.4iT - 1.48e5T^{2} \)
59 \( 1 + 354. iT - 2.05e5T^{2} \)
61 \( 1 + 595. iT - 2.26e5T^{2} \)
67 \( 1 + 31.8iT - 3.00e5T^{2} \)
71 \( 1 + 86.9T + 3.57e5T^{2} \)
73 \( 1 + 320. iT - 3.89e5T^{2} \)
79 \( 1 + 518. iT - 4.93e5T^{2} \)
83 \( 1 - 520.T + 5.71e5T^{2} \)
89 \( 1 - 40.5iT - 7.04e5T^{2} \)
97 \( 1 - 1.55e3iT - 9.12e5T^{2} \)
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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

−11.23627927562234832496803392954, −9.744295467619680809499346694491, −9.167748595931700273273781947881, −7.81437291129316614606568764974, −7.42466276344014821109799713722, −6.25096407358088160090090253896, −5.26330732104936588789336001599, −3.97061199855717042830504101431, −3.19521884644467671280776855251, −0.43822820751962221840103793161, 0.930649395274384715049159707102, 2.86583913089037374452046630368, 3.62343892018138561662282793557, 4.75336297238262469494327376004, 5.96897385635262844696667830473, 7.25420298412391938417453884558, 8.350485746419835323170776390401, 9.409182821177489214294721867707, 10.12610644010930972774319436789, 11.18352496708383230123062144467

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