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Properties

Label 2-5760-24.11-c1-0-57
Degree 22
Conductor 57605760
Sign −0.985+0.169i-0.985 + 0.169i
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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  − 5-s − 0.116i·7-s − 5.10i·11-s − 3.81i·13-s − 2.16i·17-s + 0.828·19-s + 2.39·23-s + 25-s − 5.06·29-s + 1.83i·31-s + 0.116i·35-s − 0.421i·37-s − 4.80i·41-s − 7.39·43-s − 3.88·47-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.0440i·7-s − 1.54i·11-s − 1.05i·13-s − 0.525i·17-s + 0.190·19-s + 0.499·23-s + 0.200·25-s − 0.939·29-s + 0.329i·31-s + 0.0196i·35-s − 0.0692i·37-s − 0.750i·41-s − 1.12·43-s − 0.567·47-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=((−0.985+0.169i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=((−0.985+0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{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: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: −0.985+0.169i-0.985 + 0.169i
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5760(4031,⋅)\chi_{5760} (4031, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), −0.985+0.169i)(2,\ 5760,\ (\ :1/2),\ -0.985 + 0.169i)

Particular Values

L(1)L(1) ≈\approx 0.78599351530.7859935153
L(12)L(\frac12) ≈\approx 0.78599351530.7859935153
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+T 1 + T
good7 1+0.116iT−7T2 1 + 0.116iT - 7T^{2}
11 1+5.10iT−11T2 1 + 5.10iT - 11T^{2}
13 1+3.81iT−13T2 1 + 3.81iT - 13T^{2}
17 1+2.16iT−17T2 1 + 2.16iT - 17T^{2}
19 1−0.828T+19T2 1 - 0.828T + 19T^{2}
23 1−2.39T+23T2 1 - 2.39T + 23T^{2}
29 1+5.06T+29T2 1 + 5.06T + 29T^{2}
31 1−1.83iT−31T2 1 - 1.83iT - 31T^{2}
37 1+0.421iT−37T2 1 + 0.421iT - 37T^{2}
41 1+4.80iT−41T2 1 + 4.80iT - 41T^{2}
43 1+7.39T+43T2 1 + 7.39T + 43T^{2}
47 1+3.88T+47T2 1 + 3.88T + 47T^{2}
53 1−7.39T+53T2 1 - 7.39T + 53T^{2}
59 1+7.60iT−59T2 1 + 7.60iT - 59T^{2}
61 1−1.17iT−61T2 1 - 1.17iT - 61T^{2}
67 1−3.39T+67T2 1 - 3.39T + 67T^{2}
71 1+13.0T+71T2 1 + 13.0T + 71T^{2}
73 1+7.65T+73T2 1 + 7.65T + 73T^{2}
79 1−7.49iT−79T2 1 - 7.49iT - 79T^{2}
83 1+1.09iT−83T2 1 + 1.09iT - 83T^{2}
89 1−15.6iT−89T2 1 - 15.6iT - 89T^{2}
97 1+4.06T+97T2 1 + 4.06T + 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

−7.78391711642023509613097459447, −7.18715738224646120720517522045, −6.33017264621681236500310297720, −5.51589718389375422664969875839, −5.08706026699377368898593652615, −3.87396678954188216868415586324, −3.32761575058947119148011066637, −2.57386900591019539441423804627, −1.13413327915060873349981294316, −0.21892304523687679296731591525, 1.45421150391781325594868390018, 2.19278820064574014103375060601, 3.28180869456361667953334963611, 4.23889456940082067184280152168, 4.60645814886477070042853456548, 5.56254892446515200458911732041, 6.44203439871789757920157012004, 7.21584831045744453685822899603, 7.49012878729655714447900218126, 8.528747020444721469325435817460

Graph of the ZZ-function along the critical line