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Properties

Label 2-585-5.4-c3-0-35
Degree $2$
Conductor $585$
Sign $0.655 - 0.754i$
Analytic cond. $34.5161$
Root an. cond. $5.87504$
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  + 5.17i·2-s − 18.8·4-s + (−8.43 − 7.33i)5-s + 32.8i·7-s − 56.0i·8-s + (37.9 − 43.6i)10-s − 41.5·11-s + 13i·13-s − 170.·14-s + 139.·16-s − 49.8i·17-s − 121.·19-s + (158. + 137. i)20-s − 214. i·22-s − 136. i·23-s + ⋯
L(s)  = 1  + 1.83i·2-s − 2.35·4-s + (−0.754 − 0.655i)5-s + 1.77i·7-s − 2.47i·8-s + (1.20 − 1.38i)10-s − 1.13·11-s + 0.277i·13-s − 3.24·14-s + 2.17·16-s − 0.711i·17-s − 1.46·19-s + (1.77 + 1.54i)20-s − 2.08i·22-s − 1.24i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.655 - 0.754i$
Analytic conductor: \(34.5161\)
Root analytic conductor: \(5.87504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :3/2),\ 0.655 - 0.754i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4653472352\)
\(L(\frac12)\) \(\approx\) \(0.4653472352\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (8.43 + 7.33i)T \)
13 \( 1 - 13iT \)
good2 \( 1 - 5.17iT - 8T^{2} \)
7 \( 1 - 32.8iT - 343T^{2} \)
11 \( 1 + 41.5T + 1.33e3T^{2} \)
17 \( 1 + 49.8iT - 4.91e3T^{2} \)
19 \( 1 + 121.T + 6.85e3T^{2} \)
23 \( 1 + 136. iT - 1.21e4T^{2} \)
29 \( 1 - 126.T + 2.43e4T^{2} \)
31 \( 1 - 119.T + 2.97e4T^{2} \)
37 \( 1 - 79.5iT - 5.06e4T^{2} \)
41 \( 1 + 145.T + 6.89e4T^{2} \)
43 \( 1 + 125. iT - 7.95e4T^{2} \)
47 \( 1 - 556. iT - 1.03e5T^{2} \)
53 \( 1 - 217. iT - 1.48e5T^{2} \)
59 \( 1 - 47.1T + 2.05e5T^{2} \)
61 \( 1 - 846.T + 2.26e5T^{2} \)
67 \( 1 - 285. iT - 3.00e5T^{2} \)
71 \( 1 - 275.T + 3.57e5T^{2} \)
73 \( 1 - 149. iT - 3.89e5T^{2} \)
79 \( 1 - 302.T + 4.93e5T^{2} \)
83 \( 1 + 614. iT - 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.82e3iT - 9.12e5T^{2} \)
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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

−10.00622777351235823240787436425, −8.846146354138405380655691399648, −8.556965881003661318220880982216, −7.898288504884487049196602320243, −6.74924161907895523263978389432, −5.90679477493609575637179289970, −5.02552765360141342866345543679, −4.45458284298051955748104200376, −2.63869089705955300514975969030, −0.20477922078538556294219663699, 0.77603166031079432564295655967, 2.23796874191311466084793716099, 3.46082175050699725644417665275, 3.98434359342907226275473154040, 4.95474685900843086710072504625, 6.70961943331386296995154602931, 7.82846499493363954140176761076, 8.450835059962146231385079120037, 10.05278400686189476853863934054, 10.32023522637444577057201604637

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