↑

Properties

Label 2-45-5.4-c9-0-21
Degree 22
Conductor 4545
Sign −0.540−0.841i-0.540 - 0.841i
Analytic cond. 23.176623.1766
Root an. cond. 4.814204.81420
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 30.1i·2-s − 398.·4-s + (−755. − 1.17e3i)5-s − 1.02e4i·7-s − 3.42e3i·8-s + (−3.54e4 + 2.27e4i)10-s + 9.07e4·11-s − 1.14e5i·13-s − 3.09e5·14-s − 3.07e5·16-s + 2.07e5i·17-s − 7.57e5·19-s + (3.01e5 + 4.68e5i)20-s − 2.73e6i·22-s + 9.13e5i·23-s + ⋯
L(s)  = 1  − 1.33i·2-s − 0.778·4-s + (−0.540 − 0.841i)5-s − 1.61i·7-s − 0.295i·8-s + (−1.12 + 0.720i)10-s + 1.86·11-s − 1.11i·13-s − 2.15·14-s − 1.17·16-s + 0.603i·17-s − 1.33·19-s + (0.420 + 0.655i)20-s − 2.49i·22-s + 0.680i·23-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((−0.540−0.841i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+9/2) L(s)=((−0.540−0.841i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: −0.540−0.841i-0.540 - 0.841i
Analytic conductor: 23.176623.1766
Root analytic conductor: 4.814204.81420
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ45(19,⋅)\chi_{45} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :9/2), −0.540−0.841i)(2,\ 45,\ (\ :9/2),\ -0.540 - 0.841i)

Particular Values

L(5)L(5) ≈\approx 0.754792+1.38191i0.754792 + 1.38191i
L(12)L(\frac12) ≈\approx 0.754792+1.38191i0.754792 + 1.38191i
L(112)L(\frac{11}{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)
bad3 1 1
5 1+(755.+1.17e3i)T 1 + (755. + 1.17e3i)T
good2 1+30.1iT−512T2 1 + 30.1iT - 512T^{2}
7 1+1.02e4iT−4.03e7T2 1 + 1.02e4iT - 4.03e7T^{2}
11 1−9.07e4T+2.35e9T2 1 - 9.07e4T + 2.35e9T^{2}
13 1+1.14e5iT−1.06e10T2 1 + 1.14e5iT - 1.06e10T^{2}
17 1−2.07e5iT−1.18e11T2 1 - 2.07e5iT - 1.18e11T^{2}
19 1+7.57e5T+3.22e11T2 1 + 7.57e5T + 3.22e11T^{2}
23 1−9.13e5iT−1.80e12T2 1 - 9.13e5iT - 1.80e12T^{2}
29 1−7.82e5T+1.45e13T2 1 - 7.82e5T + 1.45e13T^{2}
31 1−7.94e6T+2.64e13T2 1 - 7.94e6T + 2.64e13T^{2}
37 1−7.25e6iT−1.29e14T2 1 - 7.25e6iT - 1.29e14T^{2}
41 1−5.80e4T+3.27e14T2 1 - 5.80e4T + 3.27e14T^{2}
43 1+1.73e7iT−5.02e14T2 1 + 1.73e7iT - 5.02e14T^{2}
47 1+2.02e7iT−1.11e15T2 1 + 2.02e7iT - 1.11e15T^{2}
53 1−4.71e6iT−3.29e15T2 1 - 4.71e6iT - 3.29e15T^{2}
59 1−9.45e7T+8.66e15T2 1 - 9.45e7T + 8.66e15T^{2}
61 1+5.44e7T+1.16e16T2 1 + 5.44e7T + 1.16e16T^{2}
67 1+1.52e8iT−2.72e16T2 1 + 1.52e8iT - 2.72e16T^{2}
71 1−3.39e8T+4.58e16T2 1 - 3.39e8T + 4.58e16T^{2}
73 1−2.34e8iT−5.88e16T2 1 - 2.34e8iT - 5.88e16T^{2}
79 1+4.97e6T+1.19e17T2 1 + 4.97e6T + 1.19e17T^{2}
83 1−1.74e8iT−1.86e17T2 1 - 1.74e8iT - 1.86e17T^{2}
89 1+6.42e8T+3.50e17T2 1 + 6.42e8T + 3.50e17T^{2}
97 1+4.54e8iT−7.60e17T2 1 + 4.54e8iT - 7.60e17T^{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

−12.81884121771486838463123228598, −11.85445003447717057422894961636, −10.79812492315173055429716480443, −9.794654572409611290653912115725, −8.359449044347774362930128982708, −6.75116455664065575934296379351, −4.31140635651742738156895706066, −3.64632153983978601035844239487, −1.38874532077118551801707861388, −0.58801489903530422885681153116, 2.35608959522620515023841156039, 4.39606919536586108834765337756, 6.24425887531194310329137424844, 6.72108173973090648573801282072, 8.380824069198958270789311362832, 9.238846907971189214548425134475, 11.38826872212567852761771440589, 12.08263699095345214764194505288, 14.21852220689517025015152080975, 14.72922649197242406114831461664

Graph of the ZZ-function along the critical line