↑

Properties

Label 2-150-15.2-c7-0-18
Degree 22
Conductor 150150
Sign −0.854−0.520i-0.854 - 0.520i
Analytic cond. 46.857746.8577
Root an. cond. 6.845276.84527
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 + 5.65i)2-s + (33.2 + 32.8i)3-s − 64.0i·4-s + (−374. + 2.40i)6-s + (919. + 919. i)7-s + (362. + 362. i)8-s + (28.1 + 2.18e3i)9-s + 7.17e3i·11-s + (2.10e3 − 2.12e3i)12-s + (1.91e3 − 1.91e3i)13-s − 1.04e4·14-s − 4.09e3·16-s + (1.51e4 − 1.51e4i)17-s + (−1.25e4 − 1.22e4i)18-s − 3.02e4i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.711 + 0.702i)3-s − 0.500i·4-s + (−0.707 + 0.00455i)6-s + (1.01 + 1.01i)7-s + (0.250 + 0.250i)8-s + (0.0128 + 0.999i)9-s + 1.62i·11-s + (0.351 − 0.355i)12-s + (0.241 − 0.241i)13-s − 1.01·14-s − 0.250·16-s + (0.746 − 0.746i)17-s + (−0.506 − 0.493i)18-s − 1.01i·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((−0.854−0.520i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=((−0.854−0.520i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: −0.854−0.520i-0.854 - 0.520i
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ150(107,⋅)\chi_{150} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :7/2), −0.854−0.520i)(2,\ 150,\ (\ :7/2),\ -0.854 - 0.520i)

Particular Values

L(4)L(4) ≈\approx 2.3548262552.354826255
L(12)L(\frac12) ≈\approx 2.3548262552.354826255
L(92)L(\frac{9}{2}) 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)
bad2 1+(5.65−5.65i)T 1 + (5.65 - 5.65i)T
3 1+(−33.2−32.8i)T 1 + (-33.2 - 32.8i)T
5 1 1
good7 1+(−919.−919.i)T+8.23e5iT2 1 + (-919. - 919. i)T + 8.23e5iT^{2}
11 1−7.17e3iT−1.94e7T2 1 - 7.17e3iT - 1.94e7T^{2}
13 1+(−1.91e3+1.91e3i)T−6.27e7iT2 1 + (-1.91e3 + 1.91e3i)T - 6.27e7iT^{2}
17 1+(−1.51e4+1.51e4i)T−4.10e8iT2 1 + (-1.51e4 + 1.51e4i)T - 4.10e8iT^{2}
19 1+3.02e4iT−8.93e8T2 1 + 3.02e4iT - 8.93e8T^{2}
23 1+(−7.00e4−7.00e4i)T+3.40e9iT2 1 + (-7.00e4 - 7.00e4i)T + 3.40e9iT^{2}
29 1−6.41e4T+1.72e10T2 1 - 6.41e4T + 1.72e10T^{2}
31 1+8.21e4T+2.75e10T2 1 + 8.21e4T + 2.75e10T^{2}
37 1+(−2.28e5−2.28e5i)T+9.49e10iT2 1 + (-2.28e5 - 2.28e5i)T + 9.49e10iT^{2}
41 1−6.18e4iT−1.94e11T2 1 - 6.18e4iT - 1.94e11T^{2}
43 1+(9.28e4−9.28e4i)T−2.71e11iT2 1 + (9.28e4 - 9.28e4i)T - 2.71e11iT^{2}
47 1+(−3.90e5+3.90e5i)T−5.06e11iT2 1 + (-3.90e5 + 3.90e5i)T - 5.06e11iT^{2}
53 1+(4.52e5+4.52e5i)T+1.17e12iT2 1 + (4.52e5 + 4.52e5i)T + 1.17e12iT^{2}
59 1−1.71e5T+2.48e12T2 1 - 1.71e5T + 2.48e12T^{2}
61 1+1.71e6T+3.14e12T2 1 + 1.71e6T + 3.14e12T^{2}
67 1+(3.34e6+3.34e6i)T+6.06e12iT2 1 + (3.34e6 + 3.34e6i)T + 6.06e12iT^{2}
71 1+2.46e5iT−9.09e12T2 1 + 2.46e5iT - 9.09e12T^{2}
73 1+(1.91e6−1.91e6i)T−1.10e13iT2 1 + (1.91e6 - 1.91e6i)T - 1.10e13iT^{2}
79 1−1.41e6iT−1.92e13T2 1 - 1.41e6iT - 1.92e13T^{2}
83 1+(−4.78e6−4.78e6i)T+2.71e13iT2 1 + (-4.78e6 - 4.78e6i)T + 2.71e13iT^{2}
89 1+4.98e6T+4.42e13T2 1 + 4.98e6T + 4.42e13T^{2}
97 1+(6.44e6+6.44e6i)T+8.07e13iT2 1 + (6.44e6 + 6.44e6i)T + 8.07e13iT^{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

−11.92102436051784068056651924704, −10.90878288046566208571063530281, −9.671433595217110383281810597881, −9.107144508899977924997199896474, −8.023366237053530496264230543696, −7.18277192543086085157498734659, −5.31806809065322294962032975767, −4.68857374417458825894773120348, −2.79404839921743637207696898654, −1.57809521294185030212162951117, 0.74710606433557688490152290383, 1.48058119290040039651804238150, 3.04728959412319056476711459913, 4.11124605270616022508307577840, 6.05571791603093820008711696854, 7.43507182290627455297786478096, 8.215560119432293543916582344684, 8.919768425781274257957238928386, 10.44218463406564805473978265607, 11.12151140444186297123969952435

Graph of the ZZ-function along the critical line