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Properties

Label 2-51-3.2-c6-0-28
Degree 22
Conductor 5151
Sign −0.950−0.311i-0.950 - 0.311i
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  − 7.50i·2-s + (8.40 − 25.6i)3-s + 7.66·4-s + 24.7i·5-s + (−192. − 63.1i)6-s − 318.·7-s − 537. i·8-s + (−587. − 431. i)9-s + 185.·10-s − 711. i·11-s + (64.4 − 196. i)12-s − 791.·13-s + 2.39e3i·14-s + (633. + 207. i)15-s − 3.54e3·16-s + 1.19e3i·17-s + ⋯
L(s)  = 1  − 0.938i·2-s + (0.311 − 0.950i)3-s + 0.119·4-s + 0.197i·5-s + (−0.891 − 0.292i)6-s − 0.928·7-s − 1.05i·8-s + (−0.806 − 0.591i)9-s + 0.185·10-s − 0.534i·11-s + (0.0372 − 0.113i)12-s − 0.360·13-s + 0.871i·14-s + (0.187 + 0.0615i)15-s − 0.865·16-s + 0.242i·17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((−0.950−0.311i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+3) L(s)=((−0.950−0.311i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: −0.950−0.311i-0.950 - 0.311i
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), −0.950−0.311i)(2,\ 51,\ (\ :3),\ -0.950 - 0.311i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.234966+1.47135i0.234966 + 1.47135i
L(12)L(\frac12) ≈\approx 0.234966+1.47135i0.234966 + 1.47135i
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+(−8.40+25.6i)T 1 + (-8.40 + 25.6i)T
17 1−1.19e3iT 1 - 1.19e3iT
good2 1+7.50iT−64T2 1 + 7.50iT - 64T^{2}
5 1−24.7iT−1.56e4T2 1 - 24.7iT - 1.56e4T^{2}
7 1+318.T+1.17e5T2 1 + 318.T + 1.17e5T^{2}
11 1+711.iT−1.77e6T2 1 + 711. iT - 1.77e6T^{2}
13 1+791.T+4.82e6T2 1 + 791.T + 4.82e6T^{2}
19 1+2.30e3T+4.70e7T2 1 + 2.30e3T + 4.70e7T^{2}
23 1+1.95e3iT−1.48e8T2 1 + 1.95e3iT - 1.48e8T^{2}
29 1+3.52e3iT−5.94e8T2 1 + 3.52e3iT - 5.94e8T^{2}
31 1+1.30e3T+8.87e8T2 1 + 1.30e3T + 8.87e8T^{2}
37 1−2.61e4T+2.56e9T2 1 - 2.61e4T + 2.56e9T^{2}
41 1+6.89e4iT−4.75e9T2 1 + 6.89e4iT - 4.75e9T^{2}
43 1−1.19e5T+6.32e9T2 1 - 1.19e5T + 6.32e9T^{2}
47 1+1.88e5iT−1.07e10T2 1 + 1.88e5iT - 1.07e10T^{2}
53 1−2.62e4iT−2.21e10T2 1 - 2.62e4iT - 2.21e10T^{2}
59 1+2.19e5iT−4.21e10T2 1 + 2.19e5iT - 4.21e10T^{2}
61 1−2.52e5T+5.15e10T2 1 - 2.52e5T + 5.15e10T^{2}
67 1+6.48e4T+9.04e10T2 1 + 6.48e4T + 9.04e10T^{2}
71 1−5.09e4iT−1.28e11T2 1 - 5.09e4iT - 1.28e11T^{2}
73 1+2.37e5T+1.51e11T2 1 + 2.37e5T + 1.51e11T^{2}
79 1+6.29e5T+2.43e11T2 1 + 6.29e5T + 2.43e11T^{2}
83 1+2.50e5iT−3.26e11T2 1 + 2.50e5iT - 3.26e11T^{2}
89 1+3.90e5iT−4.96e11T2 1 + 3.90e5iT - 4.96e11T^{2}
97 1−1.40e6T+8.32e11T2 1 - 1.40e6T + 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

−13.17142285351488315187385947221, −12.52920540050799482389508589851, −11.43934414208871575750945075324, −10.23415585285898871881825855814, −8.887503433000703601263302208965, −7.22097441794737533444680760204, −6.20043391820760406171616675412, −3.45844878198169989744758538994, −2.31866509873402669062228895573, −0.59869816264911509394768018310, 2.77132483397960888952196964188, 4.64600212318102123403481114487, 6.04957977493515452296330022407, 7.43403179873349448407987139136, 8.826762825915878064173307668382, 9.921384025846937142889544173635, 11.21544547282146075015019235931, 12.74708937717364890337481382037, 14.23593740497449026071408461298, 15.06897531735356295238230503639

Graph of the ZZ-function along the critical line