Properties

Label 2-585-65.9-c1-0-18
Degree $2$
Conductor $585$
Sign $0.999 + 0.00396i$
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  + (1.02 + 0.593i)2-s + (−0.295 − 0.511i)4-s + (−1.44 + 1.71i)5-s + (1.75 − 1.01i)7-s − 3.07i·8-s + (−2.49 + 0.903i)10-s + (1.94 − 3.36i)11-s + (2.96 + 2.05i)13-s + 2.40·14-s + (1.23 − 2.14i)16-s + (4.71 − 2.72i)17-s + (2.94 + 5.09i)19-s + (1.29 + 0.231i)20-s + (3.99 − 2.30i)22-s + (0.298 + 0.172i)23-s + ⋯
L(s)  = 1  + (0.727 + 0.419i)2-s + (−0.147 − 0.255i)4-s + (−0.644 + 0.764i)5-s + (0.664 − 0.383i)7-s − 1.08i·8-s + (−0.789 + 0.285i)10-s + (0.585 − 1.01i)11-s + (0.821 + 0.570i)13-s + 0.644·14-s + (0.308 − 0.535i)16-s + (1.14 − 0.660i)17-s + (0.674 + 1.16i)19-s + (0.290 + 0.0517i)20-s + (0.850 − 0.491i)22-s + (0.0623 + 0.0359i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.999 + 0.00396i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.999 + 0.00396i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04656 - 0.00405593i\)
\(L(\frac12)\) \(\approx\) \(2.04656 - 0.00405593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 1.71i)T \)
13 \( 1 + (-2.96 - 2.05i)T \)
good2 \( 1 + (-1.02 - 0.593i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.75 + 1.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.94 + 3.36i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.71 + 2.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.94 - 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.298 - 0.172i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + (4.71 + 2.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0902 + 0.156i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.15 + 0.669i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 - 2.42iT - 53T^{2} \)
59 \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.81 - 2.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.940 + 1.62i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 7.83iT - 83T^{2} \)
89 \( 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.02 - 2.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.79081588868704006650901053921, −9.979967345737376578879884945087, −8.853344290099002411758020508332, −7.82400449408494050136796938844, −6.99144641161313004787603311853, −6.07417504724232302029895156370, −5.22800658792741331395201005386, −3.92166248092144946925227075403, −3.43708033704031881580614468253, −1.17564206107977052159223287520, 1.49442307084098923167413859699, 3.10549081509625716194294002069, 4.13114624911555259815072975768, 4.88965431517569525955397979652, 5.72628537177131329586449444854, 7.32535182407559275110890460956, 8.159301280572262365774120257877, 8.770050502722248583680405631684, 9.811024864724412956261453149504, 11.17485718344582205816837130320

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