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Properties

Label 2-585-9.4-c1-0-2
Degree $2$
Conductor $585$
Sign $-0.677 + 0.735i$
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  + (1.34 + 2.32i)2-s + (−1.55 − 0.767i)3-s + (−2.60 + 4.51i)4-s + (−0.5 + 0.866i)5-s + (−0.298 − 4.64i)6-s + (1.78 + 3.08i)7-s − 8.64·8-s + (1.82 + 2.38i)9-s − 2.68·10-s + (−0.198 − 0.343i)11-s + (7.52 − 5.01i)12-s + (0.5 − 0.866i)13-s + (−4.78 + 8.28i)14-s + (1.44 − 0.960i)15-s + (−6.39 − 11.0i)16-s − 6.55·17-s + ⋯
L(s)  = 1  + (0.949 + 1.64i)2-s + (−0.896 − 0.443i)3-s + (−1.30 + 2.25i)4-s + (−0.223 + 0.387i)5-s + (−0.121 − 1.89i)6-s + (0.673 + 1.16i)7-s − 3.05·8-s + (0.606 + 0.794i)9-s − 0.849·10-s + (−0.0598 − 0.103i)11-s + (2.17 − 1.44i)12-s + (0.138 − 0.240i)13-s + (−1.27 + 2.21i)14-s + (0.372 − 0.248i)15-s + (−1.59 − 2.76i)16-s − 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.677 + 0.735i$
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: \((2,\ 585,\ (\ :1/2),\ -0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502833 - 1.14647i\)
\(L(\frac12)\) \(\approx\) \(0.502833 - 1.14647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.55 + 0.767i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.34 - 2.32i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.78 - 3.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.198 + 0.343i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.55T + 17T^{2} \)
19 \( 1 + 0.842T + 19T^{2} \)
23 \( 1 + (-3.60 + 6.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.68 - 8.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.0513T + 37T^{2} \)
41 \( 1 + (-1.03 + 1.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.94 - 3.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.90 - 3.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 + (-0.329 + 0.570i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.12 - 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.49 - 6.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.39T + 71T^{2} \)
73 \( 1 + 0.841T + 73T^{2} \)
79 \( 1 + (-6.87 - 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.20 - 3.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (8.64 + 14.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.50443914265337394686449413166, −10.70304432657615301958780957147, −8.807933459592688829855821434428, −8.453035210651861033495949442075, −7.23189889460974614516559490758, −6.69255451935269089526597712660, −5.82018281053917424006610564322, −5.07866667485023431811394331931, −4.30596364941895316965139684467, −2.61738675378867237797122507747, 0.58603826688325966245490605498, 1.88349260225328593336616532138, 3.69920823072490824938732312164, 4.33276254580815948661056924251, 4.95535843301861744157436753772, 5.99961734844352586776919471320, 7.26399323713067266806003715573, 8.920421945361383423882120668343, 9.735428199822025783341646818084, 10.58048220974020567411940747453

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