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Properties

Label 2-360-9.4-c3-0-5
Degree $2$
Conductor $360$
Sign $0.283 - 0.958i$
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  + (−3.78 − 3.55i)3-s + (2.5 − 4.33i)5-s + (8.91 + 15.4i)7-s + (1.66 + 26.9i)9-s + (−6.05 − 10.4i)11-s + (17.1 − 29.7i)13-s + (−24.8 + 7.49i)15-s − 109.·17-s − 49.1·19-s + (21.2 − 90.2i)21-s + (−98.5 + 170. i)23-s + (−12.5 − 21.6i)25-s + (89.6 − 107. i)27-s + (119. + 207. i)29-s + (−61.2 + 106. i)31-s + ⋯
L(s)  = 1  + (−0.728 − 0.684i)3-s + (0.223 − 0.387i)5-s + (0.481 + 0.834i)7-s + (0.0616 + 0.998i)9-s + (−0.165 − 0.287i)11-s + (0.365 − 0.633i)13-s + (−0.428 + 0.129i)15-s − 1.56·17-s − 0.593·19-s + (0.220 − 0.937i)21-s + (−0.893 + 1.54i)23-s + (−0.100 − 0.173i)25-s + (0.638 − 0.769i)27-s + (0.765 + 1.32i)29-s + (−0.354 + 0.614i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.283 - 0.958i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ 0.283 - 0.958i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9168155147\)
\(L(\frac12)\) \(\approx\) \(0.9168155147\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (3.78 + 3.55i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-8.91 - 15.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (6.05 + 10.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-17.1 + 29.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 109.T + 4.91e3T^{2} \)
19 \( 1 + 49.1T + 6.85e3T^{2} \)
23 \( 1 + (98.5 - 170. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-119. - 207. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (61.2 - 106. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 277.T + 5.06e4T^{2} \)
41 \( 1 + (-191. + 332. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-192. - 334. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-230. - 398. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 484.T + 1.48e5T^{2} \)
59 \( 1 + (273. - 473. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-295. - 511. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (164. - 285. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 79.6T + 3.57e5T^{2} \)
73 \( 1 + 505.T + 3.89e5T^{2} \)
79 \( 1 + (-287. - 497. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (345. + 597. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 611.T + 7.04e5T^{2} \)
97 \( 1 + (124. + 215. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.21337005574190621989376585648, −10.62312722799674797016346729957, −9.173349366227439653589874093895, −8.391456894900881650311650684045, −7.42775881358346161419115764953, −6.16160551145015761579025882003, −5.55333596974615665765961014702, −4.46821451409704672687703461291, −2.52848662384801033249238333509, −1.32109281977479140081484268187, 0.36468862252797790990777866382, 2.24997255090826253192698939178, 4.17325627286743550963066583037, 4.51328866448338716444584242532, 6.14075339147608426088330798370, 6.69206801379056303082370226768, 8.020871472378135958920282392669, 9.172635109394330209981181094366, 10.12561092831205630227455014997, 10.87451936315112337741341694428

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