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Properties

Label 2-51-3.2-c6-0-3
Degree 22
Conductor 5151
Sign 0.483−0.875i0.483 - 0.875i
Analytic cond. 11.732711.7327
Root an. cond. 3.425313.42531
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.9i·2-s + (23.6 + 13.0i)3-s − 131.·4-s + 76.5i·5-s + (182. − 330. i)6-s − 548.·7-s + 939. i·8-s + (388. + 616. i)9-s + 1.06e3·10-s + 1.54e3i·11-s + (−3.10e3 − 1.71e3i)12-s − 3.44e3·13-s + 7.66e3i·14-s + (−998. + 1.80e3i)15-s + 4.72e3·16-s + 1.19e3i·17-s + ⋯
L(s)  = 1  − 1.74i·2-s + (0.875 + 0.483i)3-s − 2.05·4-s + 0.612i·5-s + (0.844 − 1.52i)6-s − 1.60·7-s + 1.83i·8-s + (0.532 + 0.846i)9-s + 1.06·10-s + 1.15i·11-s + (−1.79 − 0.990i)12-s − 1.56·13-s + 2.79i·14-s + (−0.295 + 0.535i)15-s + 1.15·16-s + 0.242i·17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((0.483−0.875i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+3) L(s)=((0.483−0.875i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: 0.483−0.875i0.483 - 0.875i
Analytic conductor: 11.732711.7327
Root analytic conductor: 3.425313.42531
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ51(35,⋅)\chi_{51} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 51, ( :3), 0.483−0.875i)(2,\ 51,\ (\ :3),\ 0.483 - 0.875i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.541156+0.319387i0.541156 + 0.319387i
L(12)L(\frac12) ≈\approx 0.541156+0.319387i0.541156 + 0.319387i
L(4)L(4) 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)
bad3 1+(−23.6−13.0i)T 1 + (-23.6 - 13.0i)T
17 1−1.19e3iT 1 - 1.19e3iT
good2 1+13.9iT−64T2 1 + 13.9iT - 64T^{2}
5 1−76.5iT−1.56e4T2 1 - 76.5iT - 1.56e4T^{2}
7 1+548.T+1.17e5T2 1 + 548.T + 1.17e5T^{2}
11 1−1.54e3iT−1.77e6T2 1 - 1.54e3iT - 1.77e6T^{2}
13 1+3.44e3T+4.82e6T2 1 + 3.44e3T + 4.82e6T^{2}
19 1−3.75e3T+4.70e7T2 1 - 3.75e3T + 4.70e7T^{2}
23 1+1.57e4iT−1.48e8T2 1 + 1.57e4iT - 1.48e8T^{2}
29 1+1.37e4iT−5.94e8T2 1 + 1.37e4iT - 5.94e8T^{2}
31 1+4.37e4T+8.87e8T2 1 + 4.37e4T + 8.87e8T^{2}
37 1+1.10e4T+2.56e9T2 1 + 1.10e4T + 2.56e9T^{2}
41 1−2.98e4iT−4.75e9T2 1 - 2.98e4iT - 4.75e9T^{2}
43 1+1.18e4T+6.32e9T2 1 + 1.18e4T + 6.32e9T^{2}
47 1−2.44e4iT−1.07e10T2 1 - 2.44e4iT - 1.07e10T^{2}
53 1−1.64e5iT−2.21e10T2 1 - 1.64e5iT - 2.21e10T^{2}
59 1−8.15e4iT−4.21e10T2 1 - 8.15e4iT - 4.21e10T^{2}
61 1+4.30e5T+5.15e10T2 1 + 4.30e5T + 5.15e10T^{2}
67 1+6.41e3T+9.04e10T2 1 + 6.41e3T + 9.04e10T^{2}
71 1−3.74e5iT−1.28e11T2 1 - 3.74e5iT - 1.28e11T^{2}
73 1−4.68e5T+1.51e11T2 1 - 4.68e5T + 1.51e11T^{2}
79 1+7.27e4T+2.43e11T2 1 + 7.27e4T + 2.43e11T^{2}
83 1+8.21e5iT−3.26e11T2 1 + 8.21e5iT - 3.26e11T^{2}
89 1−4.67e5iT−4.96e11T2 1 - 4.67e5iT - 4.96e11T^{2}
97 1+9.31e5T+8.32e11T2 1 + 9.31e5T + 8.32e11T^{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

−14.20675029626629769123709196755, −12.89970473136282319580942455079, −12.33663011039175065649277028506, −10.56604140470609204525602983437, −9.875506242436834135550903275608, −9.223596823861684709561195642671, −7.18626512688599366024072463254, −4.51673372308967820012274098792, −3.16319969432323171925103031665, −2.33153181746630056007258404911, 0.23717864379384447842185035480, 3.37659593993529699307039976379, 5.37881859144461047292847579639, 6.73478733353558357672993758817, 7.63182047377603294363303256548, 8.991426010886817187937782255016, 9.568835586083786566024847951630, 12.46110486311187144344843387077, 13.32672294921642420727648879438, 14.16240123005180851897589353536

Graph of the ZZ-function along the critical line