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Properties

Label 2-585-65.57-c1-0-29
Degree $2$
Conductor $585$
Sign $0.276 + 0.960i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.38·2-s − 0.0914·4-s + (2.07 − 0.843i)5-s − 3.94i·7-s − 2.88·8-s + (2.86 − 1.16i)10-s + (−3.63 − 3.63i)11-s + (2.94 − 2.08i)13-s − 5.44i·14-s − 3.80·16-s + (−2.29 + 2.29i)17-s + (1.61 + 1.61i)19-s + (−0.189 + 0.0771i)20-s + (−5.02 − 5.02i)22-s + (4.36 + 4.36i)23-s + ⋯
L(s)  = 1  + 0.976·2-s − 0.0457·4-s + (0.926 − 0.377i)5-s − 1.49i·7-s − 1.02·8-s + (0.904 − 0.368i)10-s + (−1.09 − 1.09i)11-s + (0.815 − 0.578i)13-s − 1.45i·14-s − 0.952·16-s + (−0.556 + 0.556i)17-s + (0.369 + 0.369i)19-s + (−0.0423 + 0.0172i)20-s + (−1.07 − 1.07i)22-s + (0.910 + 0.910i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.276 + 0.960i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76437 - 1.32809i\)
\(L(\frac12)\) \(\approx\) \(1.76437 - 1.32809i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 + 0.843i)T \)
13 \( 1 + (-2.94 + 2.08i)T \)
good2 \( 1 - 1.38T + 2T^{2} \)
7 \( 1 + 3.94iT - 7T^{2} \)
11 \( 1 + (3.63 + 3.63i)T + 11iT^{2} \)
17 \( 1 + (2.29 - 2.29i)T - 17iT^{2} \)
19 \( 1 + (-1.61 - 1.61i)T + 19iT^{2} \)
23 \( 1 + (-4.36 - 4.36i)T + 23iT^{2} \)
29 \( 1 - 5.30iT - 29T^{2} \)
31 \( 1 + (-6.23 + 6.23i)T - 31iT^{2} \)
37 \( 1 - 4.64iT - 37T^{2} \)
41 \( 1 + (-2.49 + 2.49i)T - 41iT^{2} \)
43 \( 1 + (0.344 + 0.344i)T + 43iT^{2} \)
47 \( 1 + 0.715iT - 47T^{2} \)
53 \( 1 + (6.16 - 6.16i)T - 53iT^{2} \)
59 \( 1 + (-4.26 + 4.26i)T - 59iT^{2} \)
61 \( 1 - 6.10T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 + (-0.129 + 0.129i)T - 71iT^{2} \)
73 \( 1 - 8.08T + 73T^{2} \)
79 \( 1 - 8.80iT - 79T^{2} \)
83 \( 1 + 8.48iT - 83T^{2} \)
89 \( 1 + (11.8 - 11.8i)T - 89iT^{2} \)
97 \( 1 - 9.42T + 97T^{2} \)
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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

−10.62650448746196357872180329231, −9.795892285547348275737384981315, −8.722097248170407456236972946607, −7.904829755884956705534330234559, −6.56241043982723932840578342085, −5.72649837330368958550555861744, −4.98997930360406758226105641231, −3.87504539707612538368523562906, −2.96201136128042868158757170757, −0.959628433803881333891728422354, 2.28937047562848921523605091690, 2.90988327377863022495116447298, 4.61960429151740863717513213228, 5.23777312428601594524224440284, 6.11321206137296962584217976058, 6.89668941402799129838689939115, 8.471594942679098005214737196263, 9.175766735466619102270860306334, 9.891794693633052298046147725068, 11.05460549711753185156087819244

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