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Properties

Label 2-585-13.9-c1-0-14
Degree $2$
Conductor $585$
Sign $0.980 - 0.196i$
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  + (−0.837 + 1.45i)2-s + (−0.403 − 0.698i)4-s + 5-s + (−1.81 − 3.15i)7-s − 2·8-s + (−0.837 + 1.45i)10-s + (0.806 − 1.39i)11-s + (3.22 − 1.61i)13-s + 6.09·14-s + (2.48 − 4.29i)16-s + (−1.64 − 2.84i)17-s + (3.55 + 6.16i)19-s + (−0.403 − 0.698i)20-s + (1.35 + 2.33i)22-s + (2.86 − 4.96i)23-s + ⋯
L(s)  = 1  + (−0.592 + 1.02i)2-s + (−0.201 − 0.349i)4-s + 0.447·5-s + (−0.687 − 1.19i)7-s − 0.707·8-s + (−0.264 + 0.458i)10-s + (0.243 − 0.420i)11-s + (0.893 − 0.448i)13-s + 1.62·14-s + (0.620 − 1.07i)16-s + (−0.398 − 0.690i)17-s + (0.816 + 1.41i)19-s + (−0.0901 − 0.156i)20-s + (0.287 + 0.498i)22-s + (0.598 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02484 + 0.101462i\)
\(L(\frac12)\) \(\approx\) \(1.02484 + 0.101462i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (-3.22 + 1.61i)T \)
good2 \( 1 + (0.837 - 1.45i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.81 + 3.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.806 + 1.39i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.64 + 2.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.55 - 6.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 + 4.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.72 + 6.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.76T + 31T^{2} \)
37 \( 1 + (-5.96 + 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.56 - 4.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.54 - 6.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.67T + 47T^{2} \)
53 \( 1 - 1.42T + 53T^{2} \)
59 \( 1 + (-5.78 - 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.70 - 4.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.91 + 3.31i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 - 2.45T + 79T^{2} \)
83 \( 1 - 2.96T + 83T^{2} \)
89 \( 1 + (-0.590 + 1.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.50 + 7.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47130930183632093542257756623, −9.673650013296591280081771326530, −8.903844479520912686525945159612, −7.920224900905711722660327476133, −7.20856873949418918021303635734, −6.30876296993635163381821979104, −5.70663071490076141156907856978, −4.07312478598554639064216369260, −2.99320014475872109041001728101, −0.77187670609570541346524471998, 1.42873053028325911166119081858, 2.56499271330492702621251030318, 3.50212815679732663165076297759, 5.19752857962159650270072640070, 6.13651798795885321277349892628, 6.97208709730623164580507266269, 8.680918239370591064741353658383, 9.091604230067039845150286966006, 9.691294128150895608675654909857, 10.71599216446648755329077256092

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