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Properties

Label 2-585-5.4-c1-0-22
Degree $2$
Conductor $585$
Sign $0.958 + 0.285i$
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  + 1.22i·2-s + 0.504·4-s + (0.638 − 2.14i)5-s − 4.18i·7-s + 3.06i·8-s + (2.62 + 0.781i)10-s − 1.89·11-s − i·13-s + 5.11·14-s − 2.73·16-s − 1.73i·17-s + 1.11·19-s + (0.322 − 1.08i)20-s − 2.32i·22-s − 9.30i·23-s + ⋯
L(s)  = 1  + 0.864i·2-s + 0.252·4-s + (0.285 − 0.958i)5-s − 1.58i·7-s + 1.08i·8-s + (0.828 + 0.247i)10-s − 0.572·11-s − 0.277i·13-s + 1.36·14-s − 0.684·16-s − 0.421i·17-s + 0.256·19-s + (0.0720 − 0.241i)20-s − 0.494i·22-s − 1.93i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64272 - 0.239639i\)
\(L(\frac12)\) \(\approx\) \(1.64272 - 0.239639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.638 + 2.14i)T \)
13 \( 1 + iT \)
good2 \( 1 - 1.22iT - 2T^{2} \)
7 \( 1 + 4.18iT - 7T^{2} \)
11 \( 1 + 1.89T + 11T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 - 1.11T + 19T^{2} \)
23 \( 1 + 9.30iT - 23T^{2} \)
29 \( 1 - 5.00T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + 8.02T + 41T^{2} \)
43 \( 1 - 1.00iT - 43T^{2} \)
47 \( 1 - 5.63iT - 47T^{2} \)
53 \( 1 + 0.174iT - 53T^{2} \)
59 \( 1 - 8.64T + 59T^{2} \)
61 \( 1 - 3.27T + 61T^{2} \)
67 \( 1 - 11.4iT - 67T^{2} \)
71 \( 1 - 5.37T + 71T^{2} \)
73 \( 1 + 4.01iT - 73T^{2} \)
79 \( 1 - 0.613T + 79T^{2} \)
83 \( 1 - 9.08iT - 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 3.85iT - 97T^{2} \)
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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

−10.40429314355451323076332889705, −9.978243861654056780158932074370, −8.379993208308161826772209063962, −8.133263860312736384493101175896, −6.96311714911396536829193256810, −6.35279616080380635233394647798, −5.04587052327634256021622156097, −4.46385562133058295521778574610, −2.73915391516672701369977679357, −0.962223865973064726679394182650, 1.91062723006129340458793816466, 2.68937447513017032981572283352, 3.58964234358430677152918280843, 5.32313818523056081029952817207, 6.16078574037972112270732129010, 7.06212044715868822341459933035, 8.174957852742418168820206253023, 9.329261380811505450997828681210, 10.00398322496883418292076138110, 10.79956094057266532099941654571

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