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Properties

Label 2-592-37.11-c1-0-13
Degree 22
Conductor 592592
Sign 0.443+0.896i0.443 + 0.896i
Analytic cond. 4.727144.72714
Root an. cond. 2.174192.17419
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  + (1.36 − 2.36i)3-s + (1.5 + 0.866i)5-s + (−1 + 1.73i)7-s + (−2.23 − 3.86i)9-s + 4.73·11-s + (−3 − 1.73i)13-s + (4.09 − 2.36i)15-s + (6.69 − 3.86i)17-s + (−1.09 − 0.633i)19-s + (2.73 + 4.73i)21-s − 4.73i·23-s + (−1 − 1.73i)25-s − 4.00·27-s + 8.66i·29-s + 1.26i·31-s + ⋯
L(s)  = 1  + (0.788 − 1.36i)3-s + (0.670 + 0.387i)5-s + (−0.377 + 0.654i)7-s + (−0.744 − 1.28i)9-s + 1.42·11-s + (−0.832 − 0.480i)13-s + (1.05 − 0.610i)15-s + (1.62 − 0.937i)17-s + (−0.251 − 0.145i)19-s + (0.596 + 1.03i)21-s − 0.986i·23-s + (−0.200 − 0.346i)25-s − 0.769·27-s + 1.60i·29-s + 0.227i·31-s + ⋯

Functional equation

Λ(s)=(592s/2 ΓC(s) L(s)=((0.443+0.896i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=((0.443+0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: 0.443+0.896i0.443 + 0.896i
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ592(529,⋅)\chi_{592} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 592, ( :1/2), 0.443+0.896i)(2,\ 592,\ (\ :1/2),\ 0.443 + 0.896i)

Particular Values

L(1)L(1) ≈\approx 1.80159−1.11879i1.80159 - 1.11879i
L(12)L(\frac12) ≈\approx 1.80159−1.11879i1.80159 - 1.11879i
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)
bad2 1 1
37 1+(5.69+2.13i)T 1 + (5.69 + 2.13i)T
good3 1+(−1.36+2.36i)T+(−1.5−2.59i)T2 1 + (-1.36 + 2.36i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.5−0.866i)T+(2.5+4.33i)T2 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2}
7 1+(1−1.73i)T+(−3.5−6.06i)T2 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2}
11 1−4.73T+11T2 1 - 4.73T + 11T^{2}
13 1+(3+1.73i)T+(6.5+11.2i)T2 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2}
17 1+(−6.69+3.86i)T+(8.5−14.7i)T2 1 + (-6.69 + 3.86i)T + (8.5 - 14.7i)T^{2}
19 1+(1.09+0.633i)T+(9.5+16.4i)T2 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2}
23 1+4.73iT−23T2 1 + 4.73iT - 23T^{2}
29 1−8.66iT−29T2 1 - 8.66iT - 29T^{2}
31 1−1.26iT−31T2 1 - 1.26iT - 31T^{2}
41 1+(4.96−8.59i)T+(−20.5−35.5i)T2 1 + (4.96 - 8.59i)T + (-20.5 - 35.5i)T^{2}
43 1−0.928iT−43T2 1 - 0.928iT - 43T^{2}
47 1+4.73T+47T2 1 + 4.73T + 47T^{2}
53 1+(1.26+2.19i)T+(−26.5+45.8i)T2 1 + (1.26 + 2.19i)T + (-26.5 + 45.8i)T^{2}
59 1+(2.19−1.26i)T+(29.5−51.0i)T2 1 + (2.19 - 1.26i)T + (29.5 - 51.0i)T^{2}
61 1+(−1.5−0.866i)T+(30.5+52.8i)T2 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2}
67 1+(5.09−8.83i)T+(−33.5−58.0i)T2 1 + (5.09 - 8.83i)T + (-33.5 - 58.0i)T^{2}
71 1+(−1.73+3i)T+(−35.5−61.4i)T2 1 + (-1.73 + 3i)T + (-35.5 - 61.4i)T^{2}
73 1−4T+73T2 1 - 4T + 73T^{2}
79 1+(−11.4−6.63i)T+(39.5+68.4i)T2 1 + (-11.4 - 6.63i)T + (39.5 + 68.4i)T^{2}
83 1+(2.83+4.90i)T+(−41.5+71.8i)T2 1 + (2.83 + 4.90i)T + (-41.5 + 71.8i)T^{2}
89 1+(5.89−3.40i)T+(44.5−77.0i)T2 1 + (5.89 - 3.40i)T + (44.5 - 77.0i)T^{2}
97 1−7.73iT−97T2 1 - 7.73iT - 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

−10.32661883506719194998945843576, −9.473944582428718355912184372610, −8.768309892268009058682854905807, −7.80170950817519754081108825748, −6.86524692381338767908854037187, −6.35849650452927331400811528648, −5.18494800891820305635030744786, −3.28875383947870717323617446545, −2.52637846742919298994091458937, −1.31319466683809854682714035615, 1.75892190641303370850394615058, 3.48851058270205280070355394743, 3.97039160964177468677306455853, 5.11565785556131063390584949370, 6.16453483572719737902115581695, 7.41458148162080071685790421824, 8.494827002453309158329776665719, 9.423338255012495276513788957363, 9.785971265707106308953896654304, 10.37393098734813912151371848150

Graph of the ZZ-function along the critical line