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Properties

Label 2-585-65.64-c1-0-13
Degree $2$
Conductor $585$
Sign $0.657 + 0.753i$
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  + 0.246·2-s − 1.93·4-s + (−2.15 + 0.608i)5-s + 2.59·7-s − 0.971·8-s + (−0.530 + 0.150i)10-s − 3.81i·11-s + (1.54 + 3.25i)13-s + 0.639·14-s + 3.63·16-s − 4.63i·17-s − 5.02i·19-s + (4.17 − 1.18i)20-s − 0.939i·22-s + 4i·23-s + ⋯
L(s)  = 1  + 0.174·2-s − 0.969·4-s + (−0.962 + 0.272i)5-s + 0.979·7-s − 0.343·8-s + (−0.167 + 0.0474i)10-s − 1.14i·11-s + (0.427 + 0.903i)13-s + 0.170·14-s + 0.909·16-s − 1.12i·17-s − 1.15i·19-s + (0.932 − 0.263i)20-s − 0.200i·22-s + 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.657 + 0.753i$
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: \((2,\ 585,\ (\ :1/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992838 - 0.451072i\)
\(L(\frac12)\) \(\approx\) \(0.992838 - 0.451072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.608i)T \)
13 \( 1 + (-1.54 - 3.25i)T \)
good2 \( 1 - 0.246T + 2T^{2} \)
7 \( 1 - 2.59T + 7T^{2} \)
11 \( 1 + 3.81iT - 11T^{2} \)
17 \( 1 + 4.63iT - 17T^{2} \)
19 \( 1 + 5.02iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 7.87T + 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + 6.40iT - 41T^{2} \)
43 \( 1 + 0.639iT - 43T^{2} \)
47 \( 1 - 3.81T + 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 + 6.24iT - 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 - 3.08T + 73T^{2} \)
79 \( 1 + 3.36T + 79T^{2} \)
83 \( 1 + 7.23T + 83T^{2} \)
89 \( 1 - 8.83iT - 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−10.87722570715548020823318907635, −9.494983978033590000689710896111, −8.748758979168819271356666299910, −8.062849053808430888745088092609, −7.15444186953401186919775436416, −5.86061646148262865529717624977, −4.74183180545785364117460531167, −4.10627934392375713195881601266, −2.90609945819445959736187370985, −0.72227736739406260464617562700, 1.30437283698218140755700925701, 3.32886793342450437273359967647, 4.43705729034987961539627656432, 4.87670674628627103262690560428, 6.14615277468195951459007291726, 7.59300248692604949960868367334, 8.264556880581166321723363154614, 8.731881973275026652813777221451, 10.14337298603645839843094599530, 10.63748394906284137453020456985

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