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Properties

Label 2-585-5.4-c3-0-12
Degree $2$
Conductor $585$
Sign $-0.999 + 0.0222i$
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  + 5.38i·2-s − 20.9·4-s + (−0.249 − 11.1i)5-s − 12.4i·7-s − 69.7i·8-s + (60.1 − 1.34i)10-s − 56.6·11-s − 13i·13-s + 67.0·14-s + 207.·16-s − 11.4i·17-s + 98.6·19-s + (5.22 + 234. i)20-s − 304. i·22-s + 138. i·23-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.62·4-s + (−0.0222 − 0.999i)5-s − 0.672i·7-s − 3.08i·8-s + (1.90 − 0.0424i)10-s − 1.55·11-s − 0.277i·13-s + 1.27·14-s + 3.24·16-s − 0.164i·17-s + 1.19·19-s + (0.0584 + 2.62i)20-s − 2.95i·22-s + 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0222i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.999 + 0.0222i$
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: \((2,\ 585,\ (\ :3/2),\ -0.999 + 0.0222i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7878521053\)
\(L(\frac12)\) \(\approx\) \(0.7878521053\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.249 + 11.1i)T \)
13 \( 1 + 13iT \)
good2 \( 1 - 5.38iT - 8T^{2} \)
7 \( 1 + 12.4iT - 343T^{2} \)
11 \( 1 + 56.6T + 1.33e3T^{2} \)
17 \( 1 + 11.4iT - 4.91e3T^{2} \)
19 \( 1 - 98.6T + 6.85e3T^{2} \)
23 \( 1 - 138. iT - 1.21e4T^{2} \)
29 \( 1 + 145.T + 2.43e4T^{2} \)
31 \( 1 - 173.T + 2.97e4T^{2} \)
37 \( 1 - 387. iT - 5.06e4T^{2} \)
41 \( 1 + 243.T + 6.89e4T^{2} \)
43 \( 1 - 158. iT - 7.95e4T^{2} \)
47 \( 1 + 76.1iT - 1.03e5T^{2} \)
53 \( 1 - 290. iT - 1.48e5T^{2} \)
59 \( 1 - 284.T + 2.05e5T^{2} \)
61 \( 1 + 253.T + 2.26e5T^{2} \)
67 \( 1 + 743. iT - 3.00e5T^{2} \)
71 \( 1 + 839.T + 3.57e5T^{2} \)
73 \( 1 - 590. iT - 3.89e5T^{2} \)
79 \( 1 - 1.34e3T + 4.93e5T^{2} \)
83 \( 1 - 86.9iT - 5.71e5T^{2} \)
89 \( 1 - 1.20e3T + 7.04e5T^{2} \)
97 \( 1 + 321. iT - 9.12e5T^{2} \)
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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

−10.23141628339539724588725625262, −9.594758233992431570027641212916, −8.640332793684403515105779605666, −7.69577532685998374912437115151, −7.54616774749412862390320506233, −6.16244576928739458735870051247, −5.20711771789789393449352208582, −4.83206844389606025774472723483, −3.50153542383056066541078330249, −0.924173094736122493935496930717, 0.30497514264879959765901464522, 2.10493121988456729862432579835, 2.71804303582553752183646456957, 3.63905043821344447989526909898, 4.90846814659451686240922214026, 5.80835591253797114005826376904, 7.43687617855829462937759867383, 8.424472578280448320873924210594, 9.348874289581486856875274833123, 10.26506269766856150771233253096

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