Properties

Label 2-585-117.103-c1-0-20
Degree $2$
Conductor $585$
Sign $0.383 - 0.923i$
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  + (0.0488 − 0.0282i)2-s + (−1.54 + 0.787i)3-s + (−0.998 + 1.72i)4-s + (−0.866 − 0.5i)5-s + (−0.0531 + 0.0819i)6-s + (3.39 − 1.96i)7-s + 0.225i·8-s + (1.76 − 2.42i)9-s − 0.0564·10-s + (1.93 − 1.11i)11-s + (0.179 − 3.45i)12-s + (−1.61 + 3.22i)13-s + (0.110 − 0.191i)14-s + (1.72 + 0.0896i)15-s + (−1.99 − 3.44i)16-s + 5.60·17-s + ⋯
L(s)  = 1  + (0.0345 − 0.0199i)2-s + (−0.890 + 0.454i)3-s + (−0.499 + 0.864i)4-s + (−0.387 − 0.223i)5-s + (−0.0217 + 0.0334i)6-s + (1.28 − 0.741i)7-s + 0.0797i·8-s + (0.586 − 0.809i)9-s − 0.0178·10-s + (0.583 − 0.337i)11-s + (0.0516 − 0.997i)12-s + (−0.448 + 0.893i)13-s + (0.0295 − 0.0512i)14-s + (0.446 + 0.0231i)15-s + (−0.497 − 0.861i)16-s + 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870035 + 0.580514i\)
\(L(\frac12)\) \(\approx\) \(0.870035 + 0.580514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.54 - 0.787i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (1.61 - 3.22i)T \)
good2 \( 1 + (-0.0488 + 0.0282i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3.39 + 1.96i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.93 + 1.11i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 - 6.06iT - 19T^{2} \)
23 \( 1 + (1.24 - 2.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.26 + 3.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.13 + 0.654i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.55iT - 37T^{2} \)
41 \( 1 + (-9.94 - 5.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.04 - 7.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.02 - 0.591i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.19T + 53T^{2} \)
59 \( 1 + (-3.81 - 2.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.24 + 2.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.7 - 6.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 16.1iT - 71T^{2} \)
73 \( 1 + 2.60iT - 73T^{2} \)
79 \( 1 + (1.96 + 3.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.81 + 5.66i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.84iT - 89T^{2} \)
97 \( 1 + (7.85 - 4.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.11600364106875490075757333757, −9.972621697972058164580448853802, −9.234361639838143607347450397467, −7.897092966163061699132337664431, −7.67345826513331351517034474555, −6.24556324504980034366835231525, −5.03549676007963967811908445082, −4.27708890056225089959880584786, −3.62281702537653631758008633158, −1.25894722171347498954520103395, 0.821091164488045883801794256434, 2.19088677265136111119123517190, 4.23234966832926157546383487405, 5.30365277572625554965853185312, 5.57360183871291858173990263572, 6.93779568654894795589037083490, 7.75640979259587033967968486038, 8.777619780133925367834794434171, 9.780131446134424728717637501864, 10.84150942316925906984426270955

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