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Properties

Label 2-585-117.25-c1-0-17
Degree $2$
Conductor $585$
Sign $0.979 + 0.199i$
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  + (−1.91 − 1.10i)2-s + (1.62 + 0.611i)3-s + (1.44 + 2.50i)4-s + (0.866 − 0.5i)5-s + (−2.42 − 2.96i)6-s + (0.916 + 0.529i)7-s − 1.98i·8-s + (2.25 + 1.98i)9-s − 2.21·10-s + (−1.05 − 0.611i)11-s + (0.812 + 4.95i)12-s + (3.38 + 1.24i)13-s + (−1.17 − 2.02i)14-s + (1.70 − 0.280i)15-s + (0.700 − 1.21i)16-s − 1.24·17-s + ⋯
L(s)  = 1  + (−1.35 − 0.782i)2-s + (0.935 + 0.353i)3-s + (0.724 + 1.25i)4-s + (0.387 − 0.223i)5-s + (−0.991 − 1.21i)6-s + (0.346 + 0.200i)7-s − 0.701i·8-s + (0.750 + 0.660i)9-s − 0.699·10-s + (−0.319 − 0.184i)11-s + (0.234 + 1.42i)12-s + (0.938 + 0.344i)13-s + (−0.313 − 0.542i)14-s + (0.441 − 0.0724i)15-s + (0.175 − 0.303i)16-s − 0.300·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.979 + 0.199i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16326 - 0.117080i\)
\(L(\frac12)\) \(\approx\) \(1.16326 - 0.117080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 0.611i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-3.38 - 1.24i)T \)
good2 \( 1 + (1.91 + 1.10i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-0.916 - 0.529i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.05 + 0.611i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 - 4.23iT - 19T^{2} \)
23 \( 1 + (0.567 + 0.983i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.73 + 3.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.71 + 0.991i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.5iT - 37T^{2} \)
41 \( 1 + (-3.01 + 1.74i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.57 - 4.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.19 - 2.42i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.53T + 53T^{2} \)
59 \( 1 + (0.233 - 0.134i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.60 + 9.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.51 + 2.60i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.36iT - 71T^{2} \)
73 \( 1 + 8.09iT - 73T^{2} \)
79 \( 1 + (-3.73 + 6.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.26 + 0.730i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + (-3.53 - 2.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36489090035536342083029176117, −9.800119015923964130553673808793, −8.951169100911938658866235529817, −8.327448142323442866088932415126, −7.79363081766793023893519988966, −6.33592623700575695952304788800, −4.88426388634837674759116573221, −3.55382201064235181119420992140, −2.39991023671719489705418549392, −1.42320571977997128519989330151, 1.09969833694829920866732244700, 2.45380504878802312967038515591, 3.94502383723560851603369496742, 5.60432620570323631897398573295, 6.74244327682867443020917057043, 7.29119056267943109658292381076, 8.214078848919756584362836402728, 8.811101281319325576145089618963, 9.525104675966161364118489675266, 10.42092958758734018905434711963

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