↑

Properties

Label 2-51-51.50-c4-0-5
Degree 22
Conductor 5151
Sign −0.732−0.680i-0.732 - 0.680i
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  + 1.93i·2-s + (−5.02 + 7.46i)3-s + 12.2·4-s + 4.62·5-s + (−14.4 − 9.71i)6-s + 58.0i·7-s + 54.6i·8-s + (−30.4 − 75.0i)9-s + 8.93i·10-s − 163.·11-s + (−61.7 + 91.5i)12-s − 81.9·13-s − 112.·14-s + (−23.2 + 34.5i)15-s + 90.8·16-s + (−44.7 + 285. i)17-s + ⋯
L(s)  = 1  + 0.482i·2-s + (−0.558 + 0.829i)3-s + 0.766·4-s + 0.185·5-s + (−0.400 − 0.269i)6-s + 1.18i·7-s + 0.853i·8-s + (−0.375 − 0.926i)9-s + 0.0893i·10-s − 1.35·11-s + (−0.428 + 0.635i)12-s − 0.484·13-s − 0.571·14-s + (−0.103 + 0.153i)15-s + 0.354·16-s + (−0.154 + 0.987i)17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((−0.732−0.680i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+2) L(s)=((−0.732−0.680i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: −0.732−0.680i-0.732 - 0.680i
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), −0.732−0.680i)(2,\ 51,\ (\ :2),\ -0.732 - 0.680i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.499808+1.27247i0.499808 + 1.27247i
L(12)L(\frac12) ≈\approx 0.499808+1.27247i0.499808 + 1.27247i
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+(5.02−7.46i)T 1 + (5.02 - 7.46i)T
17 1+(44.7−285.i)T 1 + (44.7 - 285. i)T
good2 1−1.93iT−16T2 1 - 1.93iT - 16T^{2}
5 1−4.62T+625T2 1 - 4.62T + 625T^{2}
7 1−58.0iT−2.40e3T2 1 - 58.0iT - 2.40e3T^{2}
11 1+163.T+1.46e4T2 1 + 163.T + 1.46e4T^{2}
13 1+81.9T+2.85e4T2 1 + 81.9T + 2.85e4T^{2}
19 1−524.T+1.30e5T2 1 - 524.T + 1.30e5T^{2}
23 1−767.T+2.79e5T2 1 - 767.T + 2.79e5T^{2}
29 1−462.T+7.07e5T2 1 - 462.T + 7.07e5T^{2}
31 1+637.iT−9.23e5T2 1 + 637. iT - 9.23e5T^{2}
37 1+1.17e3iT−1.87e6T2 1 + 1.17e3iT - 1.87e6T^{2}
41 1−1.29e3T+2.82e6T2 1 - 1.29e3T + 2.82e6T^{2}
43 1−1.44e3T+3.41e6T2 1 - 1.44e3T + 3.41e6T^{2}
47 1−3.32e3iT−4.87e6T2 1 - 3.32e3iT - 4.87e6T^{2}
53 1−257.iT−7.89e6T2 1 - 257. iT - 7.89e6T^{2}
59 1+4.90e3iT−1.21e7T2 1 + 4.90e3iT - 1.21e7T^{2}
61 1−3.60e3iT−1.38e7T2 1 - 3.60e3iT - 1.38e7T^{2}
67 1−1.63e3T+2.01e7T2 1 - 1.63e3T + 2.01e7T^{2}
71 1+1.21e3T+2.54e7T2 1 + 1.21e3T + 2.54e7T^{2}
73 1+7.48e3iT−2.83e7T2 1 + 7.48e3iT - 2.83e7T^{2}
79 1−2.59e3iT−3.89e7T2 1 - 2.59e3iT - 3.89e7T^{2}
83 1+4.31e3iT−4.74e7T2 1 + 4.31e3iT - 4.74e7T^{2}
89 1+1.09e3iT−6.27e7T2 1 + 1.09e3iT - 6.27e7T^{2}
97 1+4.25e3iT−8.85e7T2 1 + 4.25e3iT - 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

−15.56304062251758991485231141845, −14.55891321056566950244586522955, −12.67048020863077947191220197914, −11.57714319514804609214193493105, −10.57845642988958164908705176355, −9.257384746263001408204375015697, −7.72335579437950986092518873834, −5.99452762165832188935295619099, −5.21576139148960239604632081159, −2.72434524772730665521247204486, 0.884663034484547804391313462900, 2.75352541247958758674217804420, 5.24084859977765054488359051553, 6.99016991779222361878219784313, 7.60975296355792106812219439491, 10.01117252606116301775354552055, 10.92989027636961145212684805454, 11.89397413447151594805949320885, 13.08935598590847846731398439920, 13.87458110328952453471115056992

Graph of the ZZ-function along the critical line