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Properties

Label 2-585-65.29-c1-0-23
Degree $2$
Conductor $585$
Sign $-0.125 + 0.992i$
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  + (−2.20 + 1.27i)2-s + (2.24 − 3.88i)4-s + (0.817 − 2.08i)5-s + (−2.54 − 1.46i)7-s + 6.31i·8-s + (0.846 + 5.62i)10-s + (−0.317 − 0.550i)11-s + (3.60 + 0.0716i)13-s + 7.48·14-s + (−3.55 − 6.16i)16-s + (1.05 + 0.611i)17-s + (0.682 − 1.18i)19-s + (−6.24 − 7.83i)20-s + (1.40 + 0.808i)22-s + (−1.86 + 1.07i)23-s + ⋯
L(s)  = 1  + (−1.55 + 0.900i)2-s + (1.12 − 1.94i)4-s + (0.365 − 0.930i)5-s + (−0.961 − 0.555i)7-s + 2.23i·8-s + (0.267 + 1.78i)10-s + (−0.0957 − 0.165i)11-s + (0.999 + 0.0198i)13-s + 1.99·14-s + (−0.889 − 1.54i)16-s + (0.257 + 0.148i)17-s + (0.156 − 0.271i)19-s + (−1.39 − 1.75i)20-s + (0.298 + 0.172i)22-s + (−0.388 + 0.224i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.125 + 0.992i$
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: \((2,\ 585,\ (\ :1/2),\ -0.125 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264087 - 0.299638i\)
\(L(\frac12)\) \(\approx\) \(0.264087 - 0.299638i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.817 + 2.08i)T \)
13 \( 1 + (-3.60 - 0.0716i)T \)
good2 \( 1 + (2.20 - 1.27i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (2.54 + 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.317 + 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.05 - 0.611i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.682 + 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + (1.05 - 0.611i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.98 + 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.18 - 0.683i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.95 - 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-6.27 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.8 + 7.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.17235181902691206364484839740, −9.319499868198401740836535340975, −8.835959938811384564727554002886, −7.930616458202066438740015548982, −7.04405103937517373668392600554, −6.13235489346077769279542612888, −5.42864805045044561234459432514, −3.74667919993048902514850376913, −1.69862365798905887183173390712, −0.37296955454060633829143896227, 1.68584599691798898852558225611, 2.87450166157146223744966424214, 3.57727177497259332258716230240, 5.80545449519849992498412204152, 6.71397016311215844528179911411, 7.60309364391246117513242117664, 8.597247747602440707372029921967, 9.450923285590722863616013138784, 9.935006492626439300998563809626, 10.82888550802467557010647423224

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