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Properties

Label 2-585-13.3-c1-0-13
Degree $2$
Conductor $585$
Sign $0.859 + 0.511i$
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 − 1.73i)4-s + 5-s + (0.5 − 0.866i)7-s + (3 + 5.19i)11-s + (2.5 + 2.59i)13-s + (−1.99 − 3.46i)16-s + (2 − 3.46i)19-s + (1 − 1.73i)20-s + (−3 − 5.19i)23-s + 25-s + (−0.999 − 1.73i)28-s + (−3 − 5.19i)29-s + 5·31-s + (0.5 − 0.866i)35-s + (−1 − 1.73i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + 0.447·5-s + (0.188 − 0.327i)7-s + (0.904 + 1.56i)11-s + (0.693 + 0.720i)13-s + (−0.499 − 0.866i)16-s + (0.458 − 0.794i)19-s + (0.223 − 0.387i)20-s + (−0.625 − 1.08i)23-s + 0.200·25-s + (−0.188 − 0.327i)28-s + (−0.557 − 0.964i)29-s + 0.898·31-s + (0.0845 − 0.146i)35-s + (−0.164 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83487 - 0.504284i\)
\(L(\frac12)\) \(\approx\) \(1.83487 - 0.504284i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.56260755059863972692153469529, −9.702889871145250923174242109704, −9.252614083169208074026220082783, −7.85558090765350425233959834146, −6.67584433257157179003630504074, −6.40232625839597325705105545264, −4.99511264854779931031422214952, −4.18638496210616983215843022608, −2.35690980881093919083696393558, −1.35605156967821264572409554034, 1.55100035165527543656502168110, 3.14382143384451334832297267850, 3.75458208441060666867643746289, 5.52156313861216815334220146915, 6.15321688625486100375517154056, 7.21925409398443950937354935443, 8.368067528462360192854415895141, 8.684852461348152437677356590353, 9.940166579244293426274106700457, 10.95939909869429806902724756195

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