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Properties

Label 2-360-120.59-c3-0-26
Degree $2$
Conductor $360$
Sign $-0.159 - 0.987i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.159 - 0.987i$
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: \((2,\ 360,\ (\ :3/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.500802305\)
\(L(\frac12)\) \(\approx\) \(1.500802305\)
\(L(\frac{5}{2})\) 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

−11.00053116579437500319530489258, −9.817844324687288558011875833534, −9.567037619498021319927372271662, −8.533605334957471611624652017866, −7.31314236571847671594859372112, −6.44598945752698239536521721562, −5.70439048973388643787171447009, −4.61732385700262475228333016056, −2.99115045666752089031823054754, −1.12776212629669456352252410526, 0.69603023979718336923465249890, 2.25474971776499364592287855742, 3.20660867686634049530027991488, 4.60241661088696946416908612897, 5.82755433029920203402675030829, 6.91216282667028381032164235810, 8.260253377031036014727523054953, 9.329797023494035263010218585301, 9.806653594139754638829297687106, 10.60245805294372761271691387531

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