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Properties

Label 2-585-9.4-c1-0-10
Degree $2$
Conductor $585$
Sign $-0.918 + 0.395i$
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 + 2.34i)2-s + (1.35 + 1.07i)3-s + (−2.68 + 4.64i)4-s + (−0.5 + 0.866i)5-s + (−0.683 + 4.64i)6-s + (−1.23 − 2.13i)7-s − 9.11·8-s + (0.689 + 2.91i)9-s − 2.71·10-s + (−0.122 − 0.212i)11-s + (−8.63 + 3.42i)12-s + (0.5 − 0.866i)13-s + (3.35 − 5.80i)14-s + (−1.60 + 0.638i)15-s + (−7.00 − 12.1i)16-s + 3.50·17-s + ⋯
L(s)  = 1  + (0.959 + 1.66i)2-s + (0.784 + 0.620i)3-s + (−1.34 + 2.32i)4-s + (−0.223 + 0.387i)5-s + (−0.278 + 1.89i)6-s + (−0.466 − 0.808i)7-s − 3.22·8-s + (0.229 + 0.973i)9-s − 0.857·10-s + (−0.0369 − 0.0639i)11-s + (−2.49 + 0.988i)12-s + (0.138 − 0.240i)13-s + (0.895 − 1.55i)14-s + (−0.415 + 0.164i)15-s + (−1.75 − 3.03i)16-s + 0.849·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.918 + 0.395i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.918 + 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503004 - 2.44204i\)
\(L(\frac12)\) \(\approx\) \(0.503004 - 2.44204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.35 - 1.07i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.35 - 2.34i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.23 + 2.13i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.122 + 0.212i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.50T + 17T^{2} \)
19 \( 1 - 7.96T + 19T^{2} \)
23 \( 1 + (2.77 - 4.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.993 + 1.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.34 - 5.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.76T + 37T^{2} \)
41 \( 1 + (2.93 - 5.07i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.96 + 5.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.29 - 3.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.83T + 53T^{2} \)
59 \( 1 + (-4.26 + 7.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.27 + 9.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.37 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.58T + 71T^{2} \)
73 \( 1 + 5.64T + 73T^{2} \)
79 \( 1 + (-2.98 - 5.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.215 - 0.372i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (2.29 + 3.96i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.28497876566386912223213935253, −9.916228046297812589902374047239, −9.323749121122717536005368292887, −7.944573450553993516231729603074, −7.71670216259138111344326858656, −6.79796939163083703660790128204, −5.63946054314692664158818467424, −4.80017568579597507177688028292, −3.51101388472913955795878221601, −3.36130855907414385744767142206, 1.03848366536862666819568048905, 2.32762050742772904283061747119, 3.17811683179857895103094170946, 4.07477659610894331667616700703, 5.35125348857180421172055199369, 6.16859592730083573939309916518, 7.68174217869653942088357669337, 8.901476386648938419318922616551, 9.476998645637048246001885134047, 10.19833800012177286642713897884

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