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Properties

Label 2-585-65.29-c1-0-10
Degree $2$
Conductor $585$
Sign $0.159 - 0.987i$
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  + (−1.85 + 1.07i)2-s + (1.30 − 2.25i)4-s + (2.16 + 0.557i)5-s + (0.635 + 0.367i)7-s + 1.30i·8-s + (−4.62 + 1.28i)10-s + (0.110 + 0.190i)11-s + (−0.632 + 3.54i)13-s − 1.57·14-s + (1.20 + 2.08i)16-s + (−0.710 − 0.409i)17-s + (1.61 − 2.78i)19-s + (4.08 − 4.16i)20-s + (−0.409 − 0.236i)22-s + (6.92 − 3.99i)23-s + ⋯
L(s)  = 1  + (−1.31 + 0.758i)2-s + (0.652 − 1.12i)4-s + (0.968 + 0.249i)5-s + (0.240 + 0.138i)7-s + 0.461i·8-s + (−1.46 + 0.407i)10-s + (0.0332 + 0.0575i)11-s + (−0.175 + 0.984i)13-s − 0.421·14-s + (0.301 + 0.522i)16-s + (−0.172 − 0.0994i)17-s + (0.369 − 0.639i)19-s + (0.912 − 0.931i)20-s + (−0.0873 − 0.0504i)22-s + (1.44 − 0.833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.159 - 0.987i$
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: \((2,\ 585,\ (\ :1/2),\ 0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695116 + 0.591977i\)
\(L(\frac12)\) \(\approx\) \(0.695116 + 0.591977i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 0.557i)T \)
13 \( 1 + (0.632 - 3.54i)T \)
good2 \( 1 + (1.85 - 1.07i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.635 - 0.367i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.110 - 0.190i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.710 + 0.409i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 2.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.92 + 3.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.51 - 2.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.27T + 31T^{2} \)
37 \( 1 + (-4.44 + 2.56i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.87 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.62 - 2.66i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.80iT - 47T^{2} \)
53 \( 1 + 4.27iT - 53T^{2} \)
59 \( 1 + (1.08 - 1.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.03 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.38 - 0.797i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.86iT - 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 4.38iT - 83T^{2} \)
89 \( 1 + (2.16 + 3.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71743660914514342721257653827, −9.539252593585209124266692741586, −9.283724874252533576181793701771, −8.417960378382434847995274436154, −7.23245633361337621121344057361, −6.72502941341950522208275047113, −5.77441701955685969503440845059, −4.58940057806145469612778961704, −2.68594281465994684250669334182, −1.29116482949889025060865489090, 0.948984633139316047286101389861, 2.11305943941100959166410531826, 3.26732904375266468473662734901, 5.03551744304644396911149816573, 5.90826836155227440281387650869, 7.33464777929617104113904995995, 8.065631351760443055474180289304, 9.150503985785404398473789756839, 9.484055637408180020432998216545, 10.58379727944027251044799169912

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