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Properties

Label 2-585-65.29-c1-0-27
Degree $2$
Conductor $585$
Sign $-0.522 + 0.852i$
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  + (−1.35 + 0.783i)2-s + (0.226 − 0.391i)4-s + (1.90 − 1.16i)5-s + (−0.331 − 0.191i)7-s − 2.42i·8-s + (−1.67 + 3.07i)10-s + (−2.30 − 3.99i)11-s + (−3.53 + 0.731i)13-s + 0.598·14-s + (2.35 + 4.07i)16-s + (−4.29 − 2.47i)17-s + (−2.05 + 3.55i)19-s + (−0.0254 − 1.01i)20-s + (6.26 + 3.61i)22-s + (−5.72 + 3.30i)23-s + ⋯
L(s)  = 1  + (−0.959 + 0.553i)2-s + (0.113 − 0.195i)4-s + (0.853 − 0.521i)5-s + (−0.125 − 0.0722i)7-s − 0.856i·8-s + (−0.529 + 0.972i)10-s + (−0.695 − 1.20i)11-s + (−0.979 + 0.202i)13-s + 0.159·14-s + (0.587 + 1.01i)16-s + (−1.04 − 0.600i)17-s + (−0.470 + 0.815i)19-s + (−0.00568 − 0.226i)20-s + (1.33 + 0.770i)22-s + (−1.19 + 0.689i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.522 + 0.852i$
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: \((2,\ 585,\ (\ :1/2),\ -0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130029 - 0.232131i\)
\(L(\frac12)\) \(\approx\) \(0.130029 - 0.232131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 1.16i)T \)
13 \( 1 + (3.53 - 0.731i)T \)
good2 \( 1 + (1.35 - 0.783i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.331 + 0.191i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.30 + 3.99i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.29 + 2.47i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.05 - 3.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.72 - 3.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.80T + 31T^{2} \)
37 \( 1 + (1.44 - 0.835i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.22 + 10.7i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.10 - 4.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.71iT - 47T^{2} \)
53 \( 1 - 2.07iT - 53T^{2} \)
59 \( 1 + (-4.11 + 7.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.07 + 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 5.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.56 - 2.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 + 1.45T + 79T^{2} \)
83 \( 1 - 2.02iT - 83T^{2} \)
89 \( 1 + (-2.60 - 4.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.7 - 9.10i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.13582897943415347613040390611, −9.345748938399121935208129209726, −8.716349621365383232545877718471, −7.911621452509229038598197855235, −6.95099235081248427919860545373, −5.99318812890769307656178871702, −5.05937330423297370288148656480, −3.62864897091514501419717657861, −2.03104218043925348553685762774, −0.18713766269036030774249275240, 2.04905994270606205635659678897, 2.50201457768763717400032743613, 4.49930237081136319793952855123, 5.50261729269697288773626579721, 6.64351144768722874851511720569, 7.57894071009052807906815180942, 8.624099310463212578699359421351, 9.514305541120090298826501466744, 10.11505794761207755346209324980, 10.61449446370600969864540422430

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