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Properties

Label 2-540-5.4-c1-0-5
Degree 22
Conductor 540540
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 4.311924.31192
Root an. cond. 2.076512.07651
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  + (1.58 − 1.58i)5-s − i·7-s + 3.16·11-s + 3i·13-s − 6.32i·17-s − 3·19-s − 3.16i·23-s − 5.00i·25-s + 9.48·29-s − 2·31-s + (−1.58 − 1.58i)35-s − i·37-s + 3.16·41-s + 10i·43-s + 6.32i·47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s − 0.377i·7-s + 0.953·11-s + 0.832i·13-s − 1.53i·17-s − 0.688·19-s − 0.659i·23-s − 1.00i·25-s + 1.76·29-s − 0.359·31-s + (−0.267 − 0.267i)35-s − 0.164i·37-s + 0.493·41-s + 1.52i·43-s + 0.922i·47-s + ⋯

Functional equation

Λ(s)=(540s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(540s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 540540    =    22⋅33⋅52^{2} \cdot 3^{3} \cdot 5
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 4.311924.31192
Root analytic conductor: 2.076512.07651
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ540(109,⋅)\chi_{540} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 540, ( :1/2), 0.707+0.707i)(2,\ 540,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.53725−0.636753i1.53725 - 0.636753i
L(12)L(\frac12) ≈\approx 1.53725−0.636753i1.53725 - 0.636753i
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
3 1 1
5 1+(−1.58+1.58i)T 1 + (-1.58 + 1.58i)T
good7 1+iT−7T2 1 + iT - 7T^{2}
11 1−3.16T+11T2 1 - 3.16T + 11T^{2}
13 1−3iT−13T2 1 - 3iT - 13T^{2}
17 1+6.32iT−17T2 1 + 6.32iT - 17T^{2}
19 1+3T+19T2 1 + 3T + 19T^{2}
23 1+3.16iT−23T2 1 + 3.16iT - 23T^{2}
29 1−9.48T+29T2 1 - 9.48T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+iT−37T2 1 + iT - 37T^{2}
41 1−3.16T+41T2 1 - 3.16T + 41T^{2}
43 1−10iT−43T2 1 - 10iT - 43T^{2}
47 1−6.32iT−47T2 1 - 6.32iT - 47T^{2}
53 1+9.48iT−53T2 1 + 9.48iT - 53T^{2}
59 1+6.32T+59T2 1 + 6.32T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1+11iT−67T2 1 + 11iT - 67T^{2}
71 1+9.48T+71T2 1 + 9.48T + 71T^{2}
73 1−13iT−73T2 1 - 13iT - 73T^{2}
79 1−3T+79T2 1 - 3T + 79T^{2}
83 1−15.8iT−83T2 1 - 15.8iT - 83T^{2}
89 1+12.6T+89T2 1 + 12.6T + 89T^{2}
97 1−iT−97T2 1 - iT - 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

−10.66394558388108740158397214785, −9.602449061266310040709086096207, −9.138971882917032325124576494721, −8.180086797143006838135397086891, −6.87961662612998868504155277168, −6.25582017012099008313409589401, −4.91354576492017427015098996358, −4.21871151075220772968317566886, −2.55891320612509104968977451475, −1.11141292196279116917135978453, 1.68221706706578788417056345830, 2.98148059706059539763732947131, 4.14142567463601568673380914962, 5.64314905237052058772885131642, 6.23697560124964681076913275600, 7.19536141654616463916172647649, 8.402305423639278671659480762568, 9.133847116070031301149343471843, 10.30376404642834083141339154613, 10.61786359226985500153371548654

Graph of the ZZ-function along the critical line