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Properties

Label 2-360-120.59-c3-0-19
Degree $2$
Conductor $360$
Sign $0.297 - 0.954i$
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  + (−2.59 + 1.12i)2-s + (5.45 − 5.85i)4-s + (−6.79 − 8.88i)5-s + 14.0·7-s + (−7.54 + 21.3i)8-s + (27.6 + 15.3i)10-s + 30.3i·11-s − 63.3·13-s + (−36.4 + 15.8i)14-s + (−4.50 − 63.8i)16-s − 111.·17-s + 151.·19-s + (−89.0 − 8.67i)20-s + (−34.2 − 78.8i)22-s − 48.6i·23-s + ⋯
L(s)  = 1  + (−0.916 + 0.398i)2-s + (0.681 − 0.731i)4-s + (−0.607 − 0.794i)5-s + 0.758·7-s + (−0.333 + 0.942i)8-s + (0.873 + 0.485i)10-s + 0.833i·11-s − 1.35·13-s + (−0.695 + 0.302i)14-s + (−0.0704 − 0.997i)16-s − 1.59·17-s + 1.82·19-s + (−0.995 − 0.0970i)20-s + (−0.332 − 0.763i)22-s − 0.440i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.297 - 0.954i$
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: \((2,\ 360,\ (\ :3/2),\ 0.297 - 0.954i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8305594077\)
\(L(\frac12)\) \(\approx\) \(0.8305594077\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.59 - 1.12i)T \)
3 \( 1 \)
5 \( 1 + (6.79 + 8.88i)T \)
good7 \( 1 - 14.0T + 343T^{2} \)
11 \( 1 - 30.3iT - 1.33e3T^{2} \)
13 \( 1 + 63.3T + 2.19e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 - 151.T + 6.85e3T^{2} \)
23 \( 1 + 48.6iT - 1.21e4T^{2} \)
29 \( 1 - 140.T + 2.43e4T^{2} \)
31 \( 1 - 148. iT - 2.97e4T^{2} \)
37 \( 1 - 313.T + 5.06e4T^{2} \)
41 \( 1 - 317. iT - 6.89e4T^{2} \)
43 \( 1 + 185. iT - 7.95e4T^{2} \)
47 \( 1 - 5.81iT - 1.03e5T^{2} \)
53 \( 1 + 11.0iT - 1.48e5T^{2} \)
59 \( 1 - 487. iT - 2.05e5T^{2} \)
61 \( 1 + 615. iT - 2.26e5T^{2} \)
67 \( 1 - 1.05e3iT - 3.00e5T^{2} \)
71 \( 1 - 731.T + 3.57e5T^{2} \)
73 \( 1 - 712. iT - 3.89e5T^{2} \)
79 \( 1 - 1.18e3iT - 4.93e5T^{2} \)
83 \( 1 - 571.T + 5.71e5T^{2} \)
89 \( 1 - 210. iT - 7.04e5T^{2} \)
97 \( 1 - 796. iT - 9.12e5T^{2} \)
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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

−11.26748338104252736757680108063, −10.00393866427550475060836313303, −9.300235251723142988467648766646, −8.341544055477434968761860699701, −7.57860662120904658088935248207, −6.80635560162917508372135685091, −5.16571307787154469902498590543, −4.59957713089172980655142823485, −2.44049367470838291259523967619, −1.04428712238665914067107645997, 0.47021354710467501373002711843, 2.25433297702969725570173818453, 3.29162808716444614243972997701, 4.67573608866490447936951171408, 6.31710381951184528793687829388, 7.43278589382253042113977086248, 7.88849572244022714421324305122, 9.043685577163446333632563238899, 9.933925754922680737445473149739, 10.98955360602988024025520984503

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