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Properties

Label 2-585-13.4-c1-0-3
Degree $2$
Conductor $585$
Sign $-0.252 - 0.967i$
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  + (0.633 + 0.366i)2-s + (−0.732 − 1.26i)4-s + i·5-s + (−3.86 + 2.23i)7-s − 2.53i·8-s + (−0.366 + 0.633i)10-s + (3 + 1.73i)11-s + (0.866 + 3.5i)13-s − 3.26·14-s + (−0.535 + 0.928i)16-s + (3.36 + 5.83i)17-s + (−4.73 + 2.73i)19-s + (1.26 − 0.732i)20-s + (1.26 + 2.19i)22-s + (0.267 − 0.464i)23-s + ⋯
L(s)  = 1  + (0.448 + 0.258i)2-s + (−0.366 − 0.633i)4-s + 0.447i·5-s + (−1.46 + 0.843i)7-s − 0.896i·8-s + (−0.115 + 0.200i)10-s + (0.904 + 0.522i)11-s + (0.240 + 0.970i)13-s − 0.873·14-s + (−0.133 + 0.232i)16-s + (0.816 + 1.41i)17-s + (−1.08 + 0.626i)19-s + (0.283 − 0.163i)20-s + (0.270 + 0.468i)22-s + (0.0558 − 0.0967i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.698867 + 0.904760i\)
\(L(\frac12)\) \(\approx\) \(0.698867 + 0.904760i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-0.866 - 3.5i)T \)
good2 \( 1 + (-0.633 - 0.366i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (3.86 - 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.36 - 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.73 - 2.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.267 + 0.464i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.36 - 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.19iT - 31T^{2} \)
37 \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.56 + 2.63i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.133 + 0.232i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.196iT - 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + (6.29 - 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.7 + 6.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 + 6.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 - 2.53iT - 83T^{2} \)
89 \( 1 + (-1.09 - 0.633i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.2 - 8.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.78508696632759995993115211162, −9.929900464988579942071078192840, −9.377395310214428906164001616611, −8.516870948218544002604579743587, −6.91488295905885495707368687832, −6.28689941170250164129331699146, −5.78474543324296343759292312419, −4.24867805179668452837455584890, −3.51168538098460044253316394634, −1.83926044146137998291496237781, 0.56429376745373671365365767598, 2.93124913532012893591252927253, 3.61319501772077358232080902547, 4.59261399854636875189340891307, 5.78659016646897275023533050333, 6.84077260972325405877085074232, 7.76951335903017027681210227399, 8.831614624065921093123442637677, 9.495718873236170043935111295944, 10.45471761724605824086919436723

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