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Properties

Label 2-585-9.7-c1-0-26
Degree $2$
Conductor $585$
Sign $0.998 + 0.0605i$
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  + (0.316 − 0.548i)2-s + (−1.15 + 1.29i)3-s + (0.799 + 1.38i)4-s + (−0.5 − 0.866i)5-s + (0.343 + 1.04i)6-s + (0.896 − 1.55i)7-s + 2.27·8-s + (−0.341 − 2.98i)9-s − 0.633·10-s + (2.02 − 3.51i)11-s + (−2.71 − 0.563i)12-s + (−0.5 − 0.866i)13-s + (−0.567 − 0.983i)14-s + (1.69 + 0.352i)15-s + (−0.877 + 1.51i)16-s + 7.09·17-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)2-s + (−0.665 + 0.746i)3-s + (0.399 + 0.692i)4-s + (−0.223 − 0.387i)5-s + (0.140 + 0.425i)6-s + (0.338 − 0.586i)7-s + 0.805·8-s + (−0.113 − 0.993i)9-s − 0.200·10-s + (0.611 − 1.05i)11-s + (−0.782 − 0.162i)12-s + (−0.138 − 0.240i)13-s + (−0.151 − 0.262i)14-s + (0.437 + 0.0909i)15-s + (−0.219 + 0.379i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.998 + 0.0605i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.998 + 0.0605i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58792 - 0.0481162i\)
\(L(\frac12)\) \(\approx\) \(1.58792 - 0.0481162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 - 1.29i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.316 + 0.548i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.896 + 1.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.02 + 3.51i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
19 \( 1 + 1.65T + 19T^{2} \)
23 \( 1 + (-4.07 - 7.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.816 - 1.41i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.17 + 3.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.729T + 37T^{2} \)
41 \( 1 + (0.439 + 0.761i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.76 - 4.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.67 + 8.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.39T + 53T^{2} \)
59 \( 1 + (6.45 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.54 + 4.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.18 + 3.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.80T + 71T^{2} \)
73 \( 1 + 5.00T + 73T^{2} \)
79 \( 1 + (3.41 - 5.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.25 - 5.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.59T + 89T^{2} \)
97 \( 1 + (-6.27 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.95871933439866746846999858577, −10.03999653341968620302911538065, −9.040442923669361941562304654526, −8.004744790839102663202453426685, −7.18954969814917868501230362458, −5.92614792962436664623766509528, −4.99916616849477406534754144310, −3.81716362062277186810424130244, −3.32901519273109345242165863586, −1.15449808273147756373973414148, 1.33640235581003778042805293805, 2.49899915290145620932097608246, 4.47626462656444870594943308995, 5.38084148862989501648745712237, 6.21027948336168875461996406420, 7.05213880763875685602239212852, 7.60150047530592776379049426374, 8.862945207586033085710853705039, 10.16795637981772921591609825943, 10.67039846163865907677166882375

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