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Properties

Label 2-585-65.64-c1-0-31
Degree $2$
Conductor $585$
Sign $-0.957 + 0.288i$
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  + 1.61·2-s + 0.593·4-s + (−1.11 − 1.93i)5-s − 4.86·7-s − 2.26·8-s + (−1.79 − 3.12i)10-s − 0.989i·11-s + (−0.822 + 3.51i)13-s − 7.83·14-s − 4.83·16-s − 3.83i·17-s + 2.88i·19-s + (−0.661 − 1.14i)20-s − 1.59i·22-s − 4i·23-s + ⋯
L(s)  = 1  + 1.13·2-s + 0.296·4-s + (−0.498 − 0.866i)5-s − 1.83·7-s − 0.800·8-s + (−0.568 − 0.986i)10-s − 0.298i·11-s + (−0.228 + 0.973i)13-s − 2.09·14-s − 1.20·16-s − 0.930i·17-s + 0.662i·19-s + (−0.148 − 0.257i)20-s − 0.339i·22-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.957 + 0.288i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0815126 - 0.553713i\)
\(L(\frac12)\) \(\approx\) \(0.0815126 - 0.553713i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.93i)T \)
13 \( 1 + (0.822 - 3.51i)T \)
good2 \( 1 - 1.61T + 2T^{2} \)
7 \( 1 + 4.86T + 7T^{2} \)
11 \( 1 + 0.989iT - 11T^{2} \)
17 \( 1 + 3.83iT - 17T^{2} \)
19 \( 1 - 2.88iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 2.81T + 29T^{2} \)
31 \( 1 + 6.84iT - 31T^{2} \)
37 \( 1 + 0.334T + 37T^{2} \)
41 \( 1 + 5.85iT - 41T^{2} \)
43 \( 1 + 7.83iT - 43T^{2} \)
47 \( 1 + 0.989T + 47T^{2} \)
53 \( 1 - 7.02iT - 53T^{2} \)
59 \( 1 - 6.76iT - 59T^{2} \)
61 \( 1 + 6.64T + 61T^{2} \)
67 \( 1 + 7.34T + 67T^{2} \)
71 \( 1 + 9.91iT - 71T^{2} \)
73 \( 1 + 1.64T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 1.89iT - 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−10.24114985833138183101159448296, −9.248204185503488895213343071334, −8.865297459651313256042523469994, −7.36164690434615596405020553373, −6.39441338010103264565608653682, −5.63406858973594136672847373870, −4.46986320101360372714776255921, −3.77515748095024017928537325968, −2.72532535148245931650823414259, −0.20464206153874141013601962457, 2.95156186394459922381485905124, 3.26980582096269635555467852833, 4.37605892182370816570117436801, 5.67889270317712318882202081865, 6.44999544002312926433786291758, 7.12657895167181701319829277725, 8.411529982267919021316777361796, 9.607316055322158190891831936806, 10.21116279593823550507262458359, 11.24802798487152383406519132020

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