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Properties

Label 2-360-120.59-c1-0-3
Degree 22
Conductor 360360
Sign −0.985−0.169i-0.985 - 0.169i
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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  + (0.927 + 1.06i)2-s + (−0.280 + 1.98i)4-s + (−2.18 + 0.468i)5-s − 3.02·7-s + (−2.37 + 1.53i)8-s + (−2.52 − 1.90i)10-s + 3.62i·11-s + 1.69·13-s + (−2.80 − 3.22i)14-s + (−3.84 − 1.11i)16-s − 6.60·17-s + 5.12·19-s + (−0.313 − 4.46i)20-s + (−3.86 + 3.35i)22-s + 6.67i·23-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)2-s + (−0.140 + 0.990i)4-s + (−0.977 + 0.209i)5-s − 1.14·7-s + (−0.839 + 0.543i)8-s + (−0.799 − 0.601i)10-s + 1.09i·11-s + 0.470·13-s + (−0.748 − 0.862i)14-s + (−0.960 − 0.277i)16-s − 1.60·17-s + 1.17·19-s + (−0.0700 − 0.997i)20-s + (−0.824 + 0.716i)22-s + 1.39i·23-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((−0.985−0.169i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((−0.985−0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: −0.985−0.169i-0.985 - 0.169i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(179,⋅)\chi_{360} (179, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), −0.985−0.169i)(2,\ 360,\ (\ :1/2),\ -0.985 - 0.169i)

Particular Values

L(1)L(1) ≈\approx 0.0853911+1.00072i0.0853911 + 1.00072i
L(12)L(\frac12) ≈\approx 0.0853911+1.00072i0.0853911 + 1.00072i
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+(−0.927−1.06i)T 1 + (-0.927 - 1.06i)T
3 1 1
5 1+(2.18−0.468i)T 1 + (2.18 - 0.468i)T
good7 1+3.02T+7T2 1 + 3.02T + 7T^{2}
11 1−3.62iT−11T2 1 - 3.62iT - 11T^{2}
13 1−1.69T+13T2 1 - 1.69T + 13T^{2}
17 1+6.60T+17T2 1 + 6.60T + 17T^{2}
19 1−5.12T+19T2 1 - 5.12T + 19T^{2}
23 1−6.67iT−23T2 1 - 6.67iT - 23T^{2}
29 1−6.82T+29T2 1 - 6.82T + 29T^{2}
31 1+1.73iT−31T2 1 + 1.73iT - 31T^{2}
37 1−0.371T+37T2 1 - 0.371T + 37T^{2}
41 1−5.83iT−41T2 1 - 5.83iT - 41T^{2}
43 1−5.24iT−43T2 1 - 5.24iT - 43T^{2}
47 1−0.525iT−47T2 1 - 0.525iT - 47T^{2}
53 1+10.0iT−53T2 1 + 10.0iT - 53T^{2}
59 1−4.86iT−59T2 1 - 4.86iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−13.4iT−67T2 1 - 13.4iT - 67T^{2}
71 1+2.45T+71T2 1 + 2.45T + 71T^{2}
73 1+14.5iT−73T2 1 + 14.5iT - 73T^{2}
79 1−14.1iT−79T2 1 - 14.1iT - 79T^{2}
83 1−5.79T+83T2 1 - 5.79T + 83T^{2}
89 1+10.2iT−89T2 1 + 10.2iT - 89T^{2}
97 1−9.33iT−97T2 1 - 9.33iT - 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

−11.95123385248277604810381061731, −11.32934770240250202356394046727, −9.884250526577154754359127636971, −8.961756199444896999410084728377, −7.83523755901233362957652779477, −7.00756308083384635707393211822, −6.33318686540806826871362783010, −4.89411837356652522219958835511, −3.91398066314068413538903690294, −2.90807940082637226483979586365, 0.54853001880196575795502505780, 2.83561634224986253914388939971, 3.69866118062682136003653068504, 4.75479284056672982503204191589, 6.10426730268104657162441786048, 6.89193276134901938742028675849, 8.514045595792838479555274513275, 9.160489863808419395876491007815, 10.44131988543961343324070672308, 11.08510357753680986108080781598

Graph of the ZZ-function along the critical line