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Properties

Label 2-270-5.4-c1-0-3
Degree 22
Conductor 270270
Sign 0.974+0.223i0.974 + 0.223i
Analytic cond. 2.155962.15596
Root an. cond. 1.468311.46831
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  − i·2-s − 4-s + (2.17 + 0.5i)5-s + 4.35i·7-s + i·8-s + (0.5 − 2.17i)10-s + 4.35·11-s + 4.35·14-s + 16-s − 4i·17-s − 6·19-s + (−2.17 − 0.5i)20-s − 4.35i·22-s − 2i·23-s + (4.50 + 2.17i)25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.974 + 0.223i)5-s + 1.64i·7-s + 0.353i·8-s + (0.158 − 0.689i)10-s + 1.31·11-s + 1.16·14-s + 0.250·16-s − 0.970i·17-s − 1.37·19-s + (−0.487 − 0.111i)20-s − 0.929i·22-s − 0.417i·23-s + (0.900 + 0.435i)25-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.974+0.223i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((0.974+0.223i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.974+0.223i0.974 + 0.223i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(109,⋅)\chi_{270} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), 0.974+0.223i)(2,\ 270,\ (\ :1/2),\ 0.974 + 0.223i)

Particular Values

L(1)L(1) ≈\approx 1.42295−0.161130i1.42295 - 0.161130i
L(12)L(\frac12) ≈\approx 1.42295−0.161130i1.42295 - 0.161130i
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+iT 1 + iT
3 1 1
5 1+(−2.17−0.5i)T 1 + (-2.17 - 0.5i)T
good7 1−4.35iT−7T2 1 - 4.35iT - 7T^{2}
11 1−4.35T+11T2 1 - 4.35T + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1+6T+19T2 1 + 6T + 19T^{2}
23 1+2iT−23T2 1 + 2iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−7T+31T2 1 - 7T + 31T^{2}
37 1+8.71iT−37T2 1 + 8.71iT - 37T^{2}
41 1+8.71T+41T2 1 + 8.71T + 41T^{2}
43 1−8.71iT−43T2 1 - 8.71iT - 43T^{2}
47 1+2iT−47T2 1 + 2iT - 47T^{2}
53 1+3iT−53T2 1 + 3iT - 53T^{2}
59 1+8.71T+59T2 1 + 8.71T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1−8.71iT−67T2 1 - 8.71iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+4.35iT−73T2 1 + 4.35iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+5iT−83T2 1 + 5iT - 83T^{2}
89 1−8.71T+89T2 1 - 8.71T + 89T^{2}
97 1+4.35iT−97T2 1 + 4.35iT - 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

−11.94015517362948573116104937555, −11.05841395546642375152757259305, −9.879451865879765104275725031949, −9.146929222652268802095458709429, −8.525631392692962727035906310322, −6.63369949335041670626880174907, −5.83624473813148852273036815983, −4.63088320330289034880030398323, −2.90956297276339527698007690151, −1.89475710595397661077940748203, 1.39479614281458716766363622217, 3.79686094233131321407077484596, 4.73120116441461174560227768954, 6.31807200379814711901959686957, 6.70548239590063049571171499498, 8.053229304902307451834733415791, 9.020531779319398921644257226420, 10.07713811312692383146876162396, 10.66332091886511979844005050481, 12.11978981791208913163756420069

Graph of the ZZ-function along the critical line