Properties

Label 2-585-65.29-c1-0-19
Degree $2$
Conductor $585$
Sign $0.173 + 0.984i$
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 + (0.465 − 2.61i)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 + (−0.449 − 1.24i)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 + (0.147 − 0.826i)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 + (−0.100 − 0.277i)20-s + (−0.850 − 0.491i)22-s + (−0.0623 + 0.0359i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145565 - 0.122106i\)
\(L(\frac12)\) \(\approx\) \(0.145565 - 0.122106i\)
\(L(\frac{3}{2})\) 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

−10.18158436071831742755429937657, −9.549475792515759457647917734399, −8.865552497179629927015860626798, −7.58995597284769273858895257517, −7.02339732120502868875506396384, −6.62577727030182663289615531877, −4.65459375893594842925235050424, −3.85629849993693549631355554422, −2.57876203079263162072378718853, −0.14730469439585271047318327932, 1.31273820435065317068897950309, 2.96133333651627281707083712305, 4.25527101927818831515175506208, 5.42481657544912142339621726342, 6.24702888003597683337266505702, 7.74653138686344573099104816666, 8.468576628035327469725888563640, 9.190605453917871689989413397611, 9.847724990451877032254146097988, 10.86777233961915383488501823348

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