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Properties

Label 2-51-51.50-c4-0-14
Degree 22
Conductor 5151
Sign 0.239+0.970i0.239 + 0.970i
Analytic cond. 5.271865.27186
Root an. cond. 2.296052.29605
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.64i·2-s + (3.53 − 8.27i)3-s + 9.00·4-s + 41.0·5-s + (−21.8 − 9.35i)6-s + 79.2i·7-s − 66.1i·8-s + (−56.0 − 58.5i)9-s − 108. i·10-s + 10.0·11-s + (31.8 − 74.4i)12-s − 126.·13-s + 209.·14-s + (145. − 339. i)15-s − 30.9·16-s + (−230. + 173. i)17-s + ⋯
L(s)  = 1  − 0.661i·2-s + (0.392 − 0.919i)3-s + 0.562·4-s + 1.64·5-s + (−0.608 − 0.259i)6-s + 1.61i·7-s − 1.03i·8-s + (−0.691 − 0.722i)9-s − 1.08i·10-s + 0.0832·11-s + (0.220 − 0.517i)12-s − 0.749·13-s + 1.06·14-s + (0.645 − 1.51i)15-s − 0.120·16-s + (−0.798 + 0.601i)17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((0.239+0.970i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+2) L(s)=((0.239+0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: 0.239+0.970i0.239 + 0.970i
Analytic conductor: 5.271865.27186
Root analytic conductor: 2.296052.29605
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ51(50,⋅)\chi_{51} (50, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 51, ( :2), 0.239+0.970i)(2,\ 51,\ (\ :2),\ 0.239 + 0.970i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.83724−1.43955i1.83724 - 1.43955i
L(12)L(\frac12) ≈\approx 1.83724−1.43955i1.83724 - 1.43955i
L(3)L(3) 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+(−3.53+8.27i)T 1 + (-3.53 + 8.27i)T
17 1+(230.−173.i)T 1 + (230. - 173. i)T
good2 1+2.64iT−16T2 1 + 2.64iT - 16T^{2}
5 1−41.0T+625T2 1 - 41.0T + 625T^{2}
7 1−79.2iT−2.40e3T2 1 - 79.2iT - 2.40e3T^{2}
11 1−10.0T+1.46e4T2 1 - 10.0T + 1.46e4T^{2}
13 1+126.T+2.85e4T2 1 + 126.T + 2.85e4T^{2}
19 1+553.T+1.30e5T2 1 + 553.T + 1.30e5T^{2}
23 1−152.T+2.79e5T2 1 - 152.T + 2.79e5T^{2}
29 1+301.T+7.07e5T2 1 + 301.T + 7.07e5T^{2}
31 1−342.iT−9.23e5T2 1 - 342. iT - 9.23e5T^{2}
37 1+133.iT−1.87e6T2 1 + 133. iT - 1.87e6T^{2}
41 1−2.02e3T+2.82e6T2 1 - 2.02e3T + 2.82e6T^{2}
43 1−1.49e3T+3.41e6T2 1 - 1.49e3T + 3.41e6T^{2}
47 1+451.iT−4.87e6T2 1 + 451. iT - 4.87e6T^{2}
53 1+2.82e3iT−7.89e6T2 1 + 2.82e3iT - 7.89e6T^{2}
59 1+1.24e3iT−1.21e7T2 1 + 1.24e3iT - 1.21e7T^{2}
61 1+2.18e3iT−1.38e7T2 1 + 2.18e3iT - 1.38e7T^{2}
67 1−5.08e3T+2.01e7T2 1 - 5.08e3T + 2.01e7T^{2}
71 1+4.12e3T+2.54e7T2 1 + 4.12e3T + 2.54e7T^{2}
73 1−1.69e3iT−2.83e7T2 1 - 1.69e3iT - 2.83e7T^{2}
79 1−1.10e4iT−3.89e7T2 1 - 1.10e4iT - 3.89e7T^{2}
83 1−1.19e4iT−4.74e7T2 1 - 1.19e4iT - 4.74e7T^{2}
89 1+1.36e4iT−6.27e7T2 1 + 1.36e4iT - 6.27e7T^{2}
97 1−5.21e3iT−8.85e7T2 1 - 5.21e3iT - 8.85e7T^{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

−14.40831519672928346535141532210, −12.86350690131023732282585178148, −12.57163689960264672400935761819, −11.13211610687893879744554239814, −9.661474356904405924635177727605, −8.685289823547330837189231405165, −6.66089636375318072086582613905, −5.81313337775177441295664839260, −2.48824260897433841201703117149, −1.96970331255723460844894091607, 2.34912742108660024606512917824, 4.61651158061171551897404842301, 6.13324319266543052193340941743, 7.39833333542932859676937581632, 9.093512674854452392456070355574, 10.27612323151716740038598608350, 10.94598372704315470414556061132, 13.23567750958922079497556036025, 14.13768610914980252748609122143, 14.84737043858097954447627036469

Graph of the ZZ-function along the critical line