↑

Properties

Label 2-45-5.4-c9-0-17
Degree 22
Conductor 4545
Sign −0.818+0.574i-0.818 + 0.574i
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  − 12.2i·2-s + 362.·4-s + (−1.14e3 + 803. i)5-s + 4.34e3i·7-s − 1.06e4i·8-s + (9.81e3 + 1.39e4i)10-s − 3.67e4·11-s − 1.87e5i·13-s + 5.30e4·14-s + 5.50e4·16-s + 3.94e3i·17-s + 2.37e5·19-s + (−4.14e5 + 2.91e5i)20-s + 4.49e5i·22-s − 2.19e6i·23-s + ⋯
L(s)  = 1  − 0.540i·2-s + 0.708·4-s + (−0.818 + 0.574i)5-s + 0.683i·7-s − 0.922i·8-s + (0.310 + 0.442i)10-s − 0.756·11-s − 1.82i·13-s + 0.369·14-s + 0.210·16-s + 0.0114i·17-s + 0.417·19-s + (−0.579 + 0.407i)20-s + 0.408i·22-s − 1.63i·23-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((−0.818+0.574i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+9/2) L(s)=((−0.818+0.574i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: −0.818+0.574i-0.818 + 0.574i
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), −0.818+0.574i)(2,\ 45,\ (\ :9/2),\ -0.818 + 0.574i)

Particular Values

L(5)L(5) ≈\approx 0.319226−1.01017i0.319226 - 1.01017i
L(12)L(\frac12) ≈\approx 0.319226−1.01017i0.319226 - 1.01017i
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+(1.14e3−803.i)T 1 + (1.14e3 - 803. i)T
good2 1+12.2iT−512T2 1 + 12.2iT - 512T^{2}
7 1−4.34e3iT−4.03e7T2 1 - 4.34e3iT - 4.03e7T^{2}
11 1+3.67e4T+2.35e9T2 1 + 3.67e4T + 2.35e9T^{2}
13 1+1.87e5iT−1.06e10T2 1 + 1.87e5iT - 1.06e10T^{2}
17 1−3.94e3iT−1.18e11T2 1 - 3.94e3iT - 1.18e11T^{2}
19 1−2.37e5T+3.22e11T2 1 - 2.37e5T + 3.22e11T^{2}
23 1+2.19e6iT−1.80e12T2 1 + 2.19e6iT - 1.80e12T^{2}
29 1+6.47e6T+1.45e13T2 1 + 6.47e6T + 1.45e13T^{2}
31 1+4.92e6T+2.64e13T2 1 + 4.92e6T + 2.64e13T^{2}
37 1−6.61e6iT−1.29e14T2 1 - 6.61e6iT - 1.29e14T^{2}
41 1+2.22e7T+3.27e14T2 1 + 2.22e7T + 3.27e14T^{2}
43 1+1.25e7iT−5.02e14T2 1 + 1.25e7iT - 5.02e14T^{2}
47 1+3.42e7iT−1.11e15T2 1 + 3.42e7iT - 1.11e15T^{2}
53 1−3.09e7iT−3.29e15T2 1 - 3.09e7iT - 3.29e15T^{2}
59 1−7.50e7T+8.66e15T2 1 - 7.50e7T + 8.66e15T^{2}
61 1−1.30e7T+1.16e16T2 1 - 1.30e7T + 1.16e16T^{2}
67 1−1.37e8iT−2.72e16T2 1 - 1.37e8iT - 2.72e16T^{2}
71 1−2.12e8T+4.58e16T2 1 - 2.12e8T + 4.58e16T^{2}
73 1+2.65e8iT−5.88e16T2 1 + 2.65e8iT - 5.88e16T^{2}
79 1+2.69e8T+1.19e17T2 1 + 2.69e8T + 1.19e17T^{2}
83 1−5.22e8iT−1.86e17T2 1 - 5.22e8iT - 1.86e17T^{2}
89 1−3.29e8T+3.50e17T2 1 - 3.29e8T + 3.50e17T^{2}
97 1+1.12e9iT−7.60e17T2 1 + 1.12e9iT - 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

−12.97246813478117923553165435206, −12.12683544720310304017592701599, −10.97357946189056503730462691262, −10.21913993296446254188884956554, −8.262664387501124929008862411311, −7.15019873206511938592225170376, −5.56880080779020026025246864216, −3.43043298591612643097573338262, −2.43019326337793910592443227917, −0.33965086271740723864416505606, 1.68107266329201587971483229568, 3.76876722703005229222261280469, 5.32508528193697811554590446978, 7.04627361071236380142010749380, 7.76993717119441715558051805867, 9.287779131646571957431606092515, 11.06824266834025094016153130614, 11.76624323689540720070048311848, 13.25056169706402218679284890861, 14.51572390146767509052226174113

Graph of the ZZ-function along the critical line