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Properties

Label 2-585-117.103-c1-0-35
Degree $2$
Conductor $585$
Sign $0.998 - 0.0531i$
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  + (1.22 − 0.707i)2-s + (1.65 + 0.501i)3-s + (0.00235 − 0.00407i)4-s + (−0.866 − 0.5i)5-s + (2.38 − 0.559i)6-s + (0.00992 − 0.00573i)7-s + 2.82i·8-s + (2.49 + 1.66i)9-s − 1.41·10-s + (3.20 − 1.85i)11-s + (0.00593 − 0.00557i)12-s + (−0.209 + 3.59i)13-s + (0.00811 − 0.0140i)14-s + (−1.18 − 1.26i)15-s + (2.00 + 3.47i)16-s + 4.64·17-s + ⋯
L(s)  = 1  + (0.867 − 0.500i)2-s + (0.957 + 0.289i)3-s + (0.00117 − 0.00203i)4-s + (−0.387 − 0.223i)5-s + (0.974 − 0.228i)6-s + (0.00375 − 0.00216i)7-s + 0.998i·8-s + (0.832 + 0.553i)9-s − 0.447·10-s + (0.967 − 0.558i)11-s + (0.00171 − 0.00160i)12-s + (−0.0579 + 0.998i)13-s + (0.00216 − 0.00375i)14-s + (−0.306 − 0.326i)15-s + (0.501 + 0.868i)16-s + 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.998 - 0.0531i$
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: \((2,\ 585,\ (\ :1/2),\ 0.998 - 0.0531i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.92115 + 0.0776775i\)
\(L(\frac12)\) \(\approx\) \(2.92115 + 0.0776775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.501i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.209 - 3.59i)T \)
good2 \( 1 + (-1.22 + 0.707i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.00992 + 0.00573i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.20 + 1.85i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.64T + 17T^{2} \)
19 \( 1 + 6.04iT - 19T^{2} \)
23 \( 1 + (2.42 - 4.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.46 + 7.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.72 + 3.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.52iT - 37T^{2} \)
41 \( 1 + (-3.46 - 1.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.571 + 0.989i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.97 - 5.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.37T + 53T^{2} \)
59 \( 1 + (-1.19 - 0.688i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.33 + 7.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.37 + 4.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.79iT - 71T^{2} \)
73 \( 1 + 1.97iT - 73T^{2} \)
79 \( 1 + (0.376 + 0.652i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.33 + 4.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.56iT - 89T^{2} \)
97 \( 1 + (-0.00405 + 0.00234i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.12678069073495135542332473707, −9.536174920357574728928271261032, −9.144374661958948881761081844107, −8.085216327822088097934092999468, −7.35849807777890009926401942349, −5.90330611198711734463472329701, −4.64474994767645865770021644353, −3.93516077532721165207216181070, −3.18375779771884326663980194063, −1.86805925539658159288505342317, 1.46885865411284907671601050815, 3.33274915364800312160546061009, 3.85432347100644634039335243838, 5.10305248763208917395426089366, 6.18442498334990897802673013439, 7.10383679616488738064816713792, 7.81356236734425365176154383612, 8.819120384039128023789459543594, 9.868937747989327463679072089227, 10.40870603051841102969395450768

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