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Properties

Label 2-24e2-8.5-c1-0-9
Degree 22
Conductor 576576
Sign −0.965+0.258i-0.965 + 0.258i
Analytic cond. 4.599384.59938
Root an. cond. 2.144612.14461
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  − 3.46i·5-s − 3.46·7-s − 6·17-s + 4i·19-s − 6.92·23-s − 6.99·25-s − 3.46i·29-s + 3.46·31-s + 11.9i·35-s − 6.92i·37-s + 6·41-s − 4i·43-s − 6.92·47-s + 4.99·49-s + 3.46i·53-s + ⋯
L(s)  = 1  − 1.54i·5-s − 1.30·7-s − 1.45·17-s + 0.917i·19-s − 1.44·23-s − 1.39·25-s − 0.643i·29-s + 0.622·31-s + 2.02i·35-s − 1.13i·37-s + 0.937·41-s − 0.609i·43-s − 1.01·47-s + 0.714·49-s + 0.475i·53-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s) L(s)=((−0.965+0.258i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+1/2) L(s)=((−0.965+0.258i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: −0.965+0.258i-0.965 + 0.258i
Analytic conductor: 4.599384.59938
Root analytic conductor: 2.144612.14461
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ576(289,⋅)\chi_{576} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 576, ( :1/2), −0.965+0.258i)(2,\ 576,\ (\ :1/2),\ -0.965 + 0.258i)

Particular Values

L(1)L(1) ≈\approx 0.0714142−0.542445i0.0714142 - 0.542445i
L(12)L(\frac12) ≈\approx 0.0714142−0.542445i0.0714142 - 0.542445i
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
good5 1+3.46iT−5T2 1 + 3.46iT - 5T^{2}
7 1+3.46T+7T2 1 + 3.46T + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+6.92T+23T2 1 + 6.92T + 23T^{2}
29 1+3.46iT−29T2 1 + 3.46iT - 29T^{2}
31 1−3.46T+31T2 1 - 3.46T + 31T^{2}
37 1+6.92iT−37T2 1 + 6.92iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+6.92T+47T2 1 + 6.92T + 47T^{2}
53 1−3.46iT−53T2 1 - 3.46iT - 53T^{2}
59 1+12iT−59T2 1 + 12iT - 59T^{2}
61 1+6.92iT−61T2 1 + 6.92iT - 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1−6.92T+71T2 1 - 6.92T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−10.02989057263215053938642930455, −9.457353418372534879309284036392, −8.656571848222829156585026925487, −7.84507612092177261103405542132, −6.49884461459931999062935756134, −5.78634424990708311522391769652, −4.57785343525255765807176907092, −3.73345116502139062215289856543, −2.07069782043318117443728681898, −0.28700099775010193211999135644, 2.45087938671556506621215799675, 3.21163461637970297056113286163, 4.37100312099716250778362272705, 6.04917201706116232523680282273, 6.61726129034001713973828793813, 7.24922420199386008717698723083, 8.531454437995464666502181061694, 9.618582323950146814403088677850, 10.21986442940077278589817239545, 11.04976536597921975684010107815

Graph of the ZZ-function along the critical line