↑

Properties

Label 2-51-51.50-c4-0-2
Degree 22
Conductor 5151
Sign −0.866−0.498i-0.866 - 0.498i
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

Related objects

Downloads

Learn more

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)=((−0.866−0.498i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+2) L(s)=((−0.866−0.498i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: −0.866−0.498i-0.866 - 0.498i
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), −0.866−0.498i)(2,\ 51,\ (\ :2),\ -0.866 - 0.498i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.156724+0.586760i0.156724 + 0.586760i
L(12)L(\frac12) ≈\approx 0.156724+0.586760i0.156724 + 0.586760i
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}
show more
show less
   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

−15.33641545951758291485840457892, −14.52220380380375932430906754779, −12.33604240143802544025256810519, −12.08554185308942553768839968182, −11.05021390470931292814478483066, −8.520208328388900894671052557778, −7.75182080938957812808759525962, −6.59696191991323926186529951135, −5.23631265479648478315000201032, −2.55020241492113276374934490023, 0.36356770968680621380221856203, 3.55301197500954418434980589820, 4.33887995703699860971527737146, 6.81409953814302994250422727874, 8.022718814214034373068750363681, 10.12102344099443243482423828048, 10.76225232810275751013628870324, 11.67468693415480711398732394763, 12.64704439131479912355601625599, 14.61042499382134324011856874651

Graph of the ZZ-function along the critical line