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Properties

Label 2-585-9.4-c1-0-28
Degree $2$
Conductor $585$
Sign $0.314 + 0.949i$
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  + (0.210 + 0.363i)2-s + (−1.72 + 0.125i)3-s + (0.911 − 1.57i)4-s + (0.5 − 0.866i)5-s + (−0.408 − 0.602i)6-s + (0.421 + 0.730i)7-s + 1.60·8-s + (2.96 − 0.433i)9-s + 0.420·10-s + (−1.70 − 2.96i)11-s + (−1.37 + 2.84i)12-s + (−0.5 + 0.866i)13-s + (−0.177 + 0.307i)14-s + (−0.754 + 1.55i)15-s + (−1.48 − 2.57i)16-s − 1.27·17-s + ⋯
L(s)  = 1  + (0.148 + 0.257i)2-s + (−0.997 + 0.0725i)3-s + (0.455 − 0.789i)4-s + (0.223 − 0.387i)5-s + (−0.166 − 0.245i)6-s + (0.159 + 0.276i)7-s + 0.568·8-s + (0.989 − 0.144i)9-s + 0.132·10-s + (−0.515 − 0.892i)11-s + (−0.397 + 0.820i)12-s + (−0.138 + 0.240i)13-s + (−0.0473 + 0.0820i)14-s + (−0.194 + 0.402i)15-s + (−0.371 − 0.643i)16-s − 0.308·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.314 + 0.949i$
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: \((2,\ 585,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00733 - 0.727626i\)
\(L(\frac12)\) \(\approx\) \(1.00733 - 0.727626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.125i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.210 - 0.363i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.421 - 0.730i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.70 + 2.96i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.27T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 + (-1.66 + 2.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.41 + 5.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.42 + 2.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.40T + 37T^{2} \)
41 \( 1 + (-2.96 + 5.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.78 + 8.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.37T + 53T^{2} \)
59 \( 1 + (2.16 - 3.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.79 - 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.67 - 6.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + (3.60 + 6.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.25 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.53T + 89T^{2} \)
97 \( 1 + (0.857 + 1.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.56868072757800012868639238467, −9.897154598867729562028632881266, −8.879376072793204826194108863961, −7.63554720885787017935573453316, −6.66514107052708166646145095689, −5.77208019984843712585087279832, −5.30931721334202116172220338881, −4.23037540383963037456092882484, −2.28720888540449539832885001124, −0.77239327467422195544618937972, 1.70070860351476545994207647307, 3.06069979861842150311925394788, 4.35236830790033402651579845971, 5.26028789003628155075986040885, 6.48024646681351317751418140447, 7.28794498730038382132014248836, 7.82405123165784247856652553739, 9.365000362729797297455224655019, 10.32114747782030638052872521586, 11.00075697046508534763220960490

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