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Properties

Label 2-585-5.4-c1-0-5
Degree $2$
Conductor $585$
Sign $-0.970 - 0.241i$
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  + 1.53i·2-s − 0.369·4-s + (−0.539 + 2.17i)5-s + 1.70i·7-s + 2.51i·8-s + (−3.34 − 0.829i)10-s + 2.53·11-s − i·13-s − 2.63·14-s − 4.60·16-s + 0.921i·17-s + 0.539·19-s + (0.199 − 0.800i)20-s + 3.90i·22-s − 2.82i·23-s + ⋯
L(s)  = 1  + 1.08i·2-s − 0.184·4-s + (−0.241 + 0.970i)5-s + 0.646i·7-s + 0.887i·8-s + (−1.05 − 0.262i)10-s + 0.765·11-s − 0.277i·13-s − 0.703·14-s − 1.15·16-s + 0.223i·17-s + 0.123·19-s + (0.0445 − 0.179i)20-s + 0.833i·22-s − 0.590i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.970 - 0.241i$
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: \((2,\ 585,\ (\ :1/2),\ -0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176265 + 1.44041i\)
\(L(\frac12)\) \(\approx\) \(0.176265 + 1.44041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.539 - 2.17i)T \)
13 \( 1 + iT \)
good2 \( 1 - 1.53iT - 2T^{2} \)
7 \( 1 - 1.70iT - 7T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
17 \( 1 - 0.921iT - 17T^{2} \)
19 \( 1 - 0.539T + 19T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 - 0.879T + 31T^{2} \)
37 \( 1 - 6.04iT - 37T^{2} \)
41 \( 1 + 1.26T + 41T^{2} \)
43 \( 1 - 6.43iT - 43T^{2} \)
47 \( 1 - 5.70iT - 47T^{2} \)
53 \( 1 + 8.49iT - 53T^{2} \)
59 \( 1 + 4.72T + 59T^{2} \)
61 \( 1 - 8.04T + 61T^{2} \)
67 \( 1 + 7.86iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 1.95iT - 73T^{2} \)
79 \( 1 + 0.496T + 79T^{2} \)
83 \( 1 + 8.63iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 5.91iT - 97T^{2} \)
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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

−11.23902895022936293832419612021, −10.20155317221401667575613297191, −9.128745379504940493522064122520, −8.210603403300921664817738045380, −7.45377495889860093801350994752, −6.50986951935947122750394773900, −6.03667842235439888486501510765, −4.85239923400168241173715497957, −3.42719481109665123836523182031, −2.20532411060666277455594590042, 0.835257741726428431749845635275, 1.98745775113197713419034813024, 3.62283103810032928372709524319, 4.19878837782279341927976134552, 5.48330927181671786166445385368, 6.81497524399548867300307516372, 7.63214480166902447103490788314, 8.922253183504492193360567989715, 9.489511426959401974113260747784, 10.41047226916215726576675220516

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