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Properties

Label 2-1080-5.4-c1-0-9
Degree 22
Conductor 10801080
Sign 0.994+0.100i0.994 + 0.100i
Analytic cond. 8.623848.62384
Root an. cond. 2.936632.93663
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  + (−2.22 − 0.224i)5-s + i·7-s − 0.449·11-s − 3.89i·13-s + 4.89i·17-s + 5.89·19-s − 4.44i·23-s + (4.89 + i)25-s + 0.449·29-s + 6·31-s + (0.224 − 2.22i)35-s + 0.101i·37-s + 9.34·41-s + 6i·43-s − 4.89i·47-s + ⋯
L(s)  = 1  + (−0.994 − 0.100i)5-s + 0.377i·7-s − 0.135·11-s − 1.08i·13-s + 1.18i·17-s + 1.35·19-s − 0.927i·23-s + (0.979 + 0.200i)25-s + 0.0834·29-s + 1.07·31-s + (0.0379 − 0.376i)35-s + 0.0166i·37-s + 1.45·41-s + 0.914i·43-s − 0.714i·47-s + ⋯

Functional equation

Λ(s)=(1080s/2 ΓC(s) L(s)=((0.994+0.100i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+1/2) L(s)=((0.994+0.100i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10801080    =    23⋅33⋅52^{3} \cdot 3^{3} \cdot 5
Sign: 0.994+0.100i0.994 + 0.100i
Analytic conductor: 8.623848.62384
Root analytic conductor: 2.936632.93663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1080(649,⋅)\chi_{1080} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1080, ( :1/2), 0.994+0.100i)(2,\ 1080,\ (\ :1/2),\ 0.994 + 0.100i)

Particular Values

L(1)L(1) ≈\approx 1.3188362841.318836284
L(12)L(\frac12) ≈\approx 1.3188362841.318836284
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+(2.22+0.224i)T 1 + (2.22 + 0.224i)T
good7 1−iT−7T2 1 - iT - 7T^{2}
11 1+0.449T+11T2 1 + 0.449T + 11T^{2}
13 1+3.89iT−13T2 1 + 3.89iT - 13T^{2}
17 1−4.89iT−17T2 1 - 4.89iT - 17T^{2}
19 1−5.89T+19T2 1 - 5.89T + 19T^{2}
23 1+4.44iT−23T2 1 + 4.44iT - 23T^{2}
29 1−0.449T+29T2 1 - 0.449T + 29T^{2}
31 1−6T+31T2 1 - 6T + 31T^{2}
37 1−0.101iT−37T2 1 - 0.101iT - 37T^{2}
41 1−9.34T+41T2 1 - 9.34T + 41T^{2}
43 1−6iT−43T2 1 - 6iT - 43T^{2}
47 1+4.89iT−47T2 1 + 4.89iT - 47T^{2}
53 1−4.44iT−53T2 1 - 4.44iT - 53T^{2}
59 1+4.89T+59T2 1 + 4.89T + 59T^{2}
61 1−8.79T+61T2 1 - 8.79T + 61T^{2}
67 1+14.7iT−67T2 1 + 14.7iT - 67T^{2}
71 1−11.5T+71T2 1 - 11.5T + 71T^{2}
73 1−3.89iT−73T2 1 - 3.89iT - 73T^{2}
79 1−3.89T+79T2 1 - 3.89T + 79T^{2}
83 1+7.55iT−83T2 1 + 7.55iT - 83T^{2}
89 1+12T+89T2 1 + 12T + 89T^{2}
97 1−15.8iT−97T2 1 - 15.8iT - 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

−9.939764983358776603369532663247, −8.889457253960636705652971547551, −8.100066048369795637886686014650, −7.63672542566775209068636803105, −6.48701999791385397793303490252, −5.55751772577954270718501211801, −4.61215967172612307793336079584, −3.59073139995616428503221657645, −2.66163040182725667415818103655, −0.867057937735633417282932925047, 0.928965721584992972493163018449, 2.66933529378848110097223819157, 3.72285226185615309856788897395, 4.55732187720825359557970819878, 5.50581794631919615383208040384, 6.85116609336243496825050146052, 7.33030441075050857515636205375, 8.107239784601655893503006548854, 9.163407510001933213351483462551, 9.763403724311856000301787135332

Graph of the ZZ-function along the critical line