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Properties

Label 2-585-5.4-c1-0-5
Degree 22
Conductor 585585
Sign −0.970−0.241i-0.970 - 0.241i
Analytic cond. 4.671244.67124
Root an. cond. 2.161302.16130
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(585s/2 ΓC(s) L(s)=((−0.970−0.241i) Λ‾(2−s)\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}
Λ(s)=(585s/2 ΓC(s+1/2) L(s)=((−0.970−0.241i) Λ‾(1−s)\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: 22
Conductor: 585585    =    32⋅5⋅133^{2} \cdot 5 \cdot 13
Sign: −0.970−0.241i-0.970 - 0.241i
Analytic conductor: 4.671244.67124
Root analytic conductor: 2.161302.16130
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ585(469,⋅)\chi_{585} (469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 585, ( :1/2), −0.970−0.241i)(2,\ 585,\ (\ :1/2),\ -0.970 - 0.241i)

Particular Values

L(1)L(1) ≈\approx 0.176265+1.44041i0.176265 + 1.44041i
L(12)L(\frac12) ≈\approx 0.176265+1.44041i0.176265 + 1.44041i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1+(0.539−2.17i)T 1 + (0.539 - 2.17i)T
13 1+iT 1 + iT
good2 1−1.53iT−2T2 1 - 1.53iT - 2T^{2}
7 1−1.70iT−7T2 1 - 1.70iT - 7T^{2}
11 1−2.53T+11T2 1 - 2.53T + 11T^{2}
17 1−0.921iT−17T2 1 - 0.921iT - 17T^{2}
19 1−0.539T+19T2 1 - 0.539T + 19T^{2}
23 1+2.82iT−23T2 1 + 2.82iT - 23T^{2}
29 1+5.12T+29T2 1 + 5.12T + 29T^{2}
31 1−0.879T+31T2 1 - 0.879T + 31T^{2}
37 1−6.04iT−37T2 1 - 6.04iT - 37T^{2}
41 1+1.26T+41T2 1 + 1.26T + 41T^{2}
43 1−6.43iT−43T2 1 - 6.43iT - 43T^{2}
47 1−5.70iT−47T2 1 - 5.70iT - 47T^{2}
53 1+8.49iT−53T2 1 + 8.49iT - 53T^{2}
59 1+4.72T+59T2 1 + 4.72T + 59T^{2}
61 1−8.04T+61T2 1 - 8.04T + 61T^{2}
67 1+7.86iT−67T2 1 + 7.86iT - 67T^{2}
71 1−14.4T+71T2 1 - 14.4T + 71T^{2}
73 1+1.95iT−73T2 1 + 1.95iT - 73T^{2}
79 1+0.496T+79T2 1 + 0.496T + 79T^{2}
83 1+8.63iT−83T2 1 + 8.63iT - 83T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1+5.91iT−97T2 1 + 5.91iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line