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Properties

Label 2-585-65.9-c1-0-21
Degree $2$
Conductor $585$
Sign $-0.997 + 0.0762i$
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  + (−1.35 − 0.783i)2-s + (0.226 + 0.391i)4-s + (−1.90 + 1.16i)5-s + (0.331 − 0.191i)7-s + 2.42i·8-s + (3.50 − 0.0880i)10-s + (2.30 − 3.99i)11-s + (3.53 + 0.731i)13-s − 0.598·14-s + (2.35 − 4.07i)16-s + (−4.29 + 2.47i)17-s + (−2.05 − 3.55i)19-s + (−0.888 − 0.483i)20-s + (−6.26 + 3.61i)22-s + (−5.72 − 3.30i)23-s + ⋯
L(s)  = 1  + (−0.959 − 0.553i)2-s + (0.113 + 0.195i)4-s + (−0.853 + 0.521i)5-s + (0.125 − 0.0722i)7-s + 0.856i·8-s + (1.10 − 0.0278i)10-s + (0.695 − 1.20i)11-s + (0.979 + 0.202i)13-s − 0.159·14-s + (0.587 − 1.01i)16-s + (−1.04 + 0.600i)17-s + (−0.470 − 0.815i)19-s + (−0.198 − 0.108i)20-s + (−1.33 + 0.770i)22-s + (−1.19 − 0.689i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.997 + 0.0762i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.997 + 0.0762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0104628 - 0.274183i\)
\(L(\frac12)\) \(\approx\) \(0.0104628 - 0.274183i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.90 - 1.16i)T \)
13 \( 1 + (-3.53 - 0.731i)T \)
good2 \( 1 + (1.35 + 0.783i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-0.331 + 0.191i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.30 + 3.99i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.29 - 2.47i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.05 + 3.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.72 + 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.65 - 4.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.80T + 31T^{2} \)
37 \( 1 + (-1.44 - 0.835i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.22 + 10.7i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.10 - 4.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.71iT - 47T^{2} \)
53 \( 1 + 2.07iT - 53T^{2} \)
59 \( 1 + (4.11 + 7.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.07 - 1.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.56 - 2.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 + 1.45T + 79T^{2} \)
83 \( 1 + 2.02iT - 83T^{2} \)
89 \( 1 + (2.60 - 4.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.7 - 9.10i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.65005656573804870592331756898, −9.179509347828392650364978221491, −8.700908659668441233720015949517, −7.981841790776221242413072748718, −6.76596421423624238785413829111, −5.87357784108783039683802457252, −4.31762892583537443134196725281, −3.34655420387824710745313365974, −1.85746861710128448514313015621, −0.21874053838106304712147533808, 1.59706537978504086803786065596, 3.77104311375664227627251046401, 4.37619376889478485873184364657, 5.92423464760068019022736897743, 6.99140667909027683730616939640, 7.72071598274437113634292536425, 8.442365824875431466767767655948, 9.226180877731073910172570725097, 9.885683438132363713964786469538, 11.10391658739647751390345449462

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