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Properties

Label 2-624-12.11-c3-0-7
Degree 22
Conductor 624624
Sign −0.618+0.785i-0.618 + 0.785i
Analytic cond. 36.817136.8171
Root an. cond. 6.067716.06771
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.92 + 4.82i)3-s + 18.6i·5-s − 4.79i·7-s + (−19.5 + 18.5i)9-s − 18.4·11-s − 13·13-s + (−89.9 + 35.9i)15-s + 42.4i·17-s − 58.4i·19-s + (23.1 − 9.23i)21-s − 20.6·23-s − 222.·25-s + (−127. − 58.6i)27-s − 65.8i·29-s + 129. i·31-s + ⋯
L(s)  = 1  + (0.370 + 0.928i)3-s + 1.66i·5-s − 0.258i·7-s + (−0.725 + 0.688i)9-s − 0.505·11-s − 0.277·13-s + (−1.54 + 0.618i)15-s + 0.605i·17-s − 0.705i·19-s + (0.240 − 0.0959i)21-s − 0.186·23-s − 1.78·25-s + (−0.908 − 0.417i)27-s − 0.421i·29-s + 0.748i·31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.618+0.785i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((−0.618+0.785i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.618+0.785i-0.618 + 0.785i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(287,⋅)\chi_{624} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), −0.618+0.785i)(2,\ 624,\ (\ :3/2),\ -0.618 + 0.785i)

Particular Values

L(2)L(2) ≈\approx 0.90163052000.9016305200
L(12)L(\frac12) ≈\approx 0.90163052000.9016305200
L(52)L(\frac{5}{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+(−1.92−4.82i)T 1 + (-1.92 - 4.82i)T
13 1+13T 1 + 13T
good5 1−18.6iT−125T2 1 - 18.6iT - 125T^{2}
7 1+4.79iT−343T2 1 + 4.79iT - 343T^{2}
11 1+18.4T+1.33e3T2 1 + 18.4T + 1.33e3T^{2}
17 1−42.4iT−4.91e3T2 1 - 42.4iT - 4.91e3T^{2}
19 1+58.4iT−6.85e3T2 1 + 58.4iT - 6.85e3T^{2}
23 1+20.6T+1.21e4T2 1 + 20.6T + 1.21e4T^{2}
29 1+65.8iT−2.43e4T2 1 + 65.8iT - 2.43e4T^{2}
31 1−129.iT−2.97e4T2 1 - 129. iT - 2.97e4T^{2}
37 1+280.T+5.06e4T2 1 + 280.T + 5.06e4T^{2}
41 1+49.8iT−6.89e4T2 1 + 49.8iT - 6.89e4T^{2}
43 1−336.iT−7.95e4T2 1 - 336. iT - 7.95e4T^{2}
47 1−482.T+1.03e5T2 1 - 482.T + 1.03e5T^{2}
53 1+256.iT−1.48e5T2 1 + 256. iT - 1.48e5T^{2}
59 1+96.1T+2.05e5T2 1 + 96.1T + 2.05e5T^{2}
61 1+275.T+2.26e5T2 1 + 275.T + 2.26e5T^{2}
67 1+912.iT−3.00e5T2 1 + 912. iT - 3.00e5T^{2}
71 1−842.T+3.57e5T2 1 - 842.T + 3.57e5T^{2}
73 1+851.T+3.89e5T2 1 + 851.T + 3.89e5T^{2}
79 1−305.iT−4.93e5T2 1 - 305. iT - 4.93e5T^{2}
83 1−394.T+5.71e5T2 1 - 394.T + 5.71e5T^{2}
89 1−1.35e3iT−7.04e5T2 1 - 1.35e3iT - 7.04e5T^{2}
97 1−261.T+9.12e5T2 1 - 261.T + 9.12e5T^{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

−10.65399811953720297990789748261, −10.13684724437901491152706601960, −9.200695877800232147522541947602, −8.096879101250444998255460588080, −7.26613625094839999110783026361, −6.34293507908729015909822428809, −5.22733193060623562613784522440, −4.02340282563261573850654717209, −3.12892802222378269550616620052, −2.29725216455347700948704489997, 0.24119068026179436863371928621, 1.36002558899392762670704675039, 2.47264953233651532728917553068, 3.95972915843902528067601770078, 5.19756922026556717137233442521, 5.83099754869477433387687138390, 7.19590319017204658363999993188, 7.960831720387774169335902692398, 8.771053876962845925658829989565, 9.264373332141908730667928948275

Graph of the ZZ-function along the critical line