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Properties

Label 2-1920-1920.1229-c0-0-0
Degree 22
Conductor 19201920
Sign −0.427−0.903i-0.427 - 0.903i
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.634 + 0.773i)2-s + (−0.956 + 0.290i)3-s + (−0.195 + 0.980i)4-s + (0.995 − 0.0980i)5-s + (−0.831 − 0.555i)6-s + (−0.881 + 0.471i)8-s + (0.831 − 0.555i)9-s + (0.707 + 0.707i)10-s + (−0.0980 − 0.995i)12-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)16-s + (−0.181 − 0.0750i)17-s + (0.956 + 0.290i)18-s + (1.11 + 1.36i)19-s + (−0.0980 + 0.995i)20-s + ⋯
L(s)  = 1  + (0.634 + 0.773i)2-s + (−0.956 + 0.290i)3-s + (−0.195 + 0.980i)4-s + (0.995 − 0.0980i)5-s + (−0.831 − 0.555i)6-s + (−0.881 + 0.471i)8-s + (0.831 − 0.555i)9-s + (0.707 + 0.707i)10-s + (−0.0980 − 0.995i)12-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)16-s + (−0.181 − 0.0750i)17-s + (0.956 + 0.290i)18-s + (1.11 + 1.36i)19-s + (−0.0980 + 0.995i)20-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.427−0.903i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.427−0.903i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: −0.427−0.903i-0.427 - 0.903i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1920(1229,⋅)\chi_{1920} (1229, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :0), −0.427−0.903i)(2,\ 1920,\ (\ :0),\ -0.427 - 0.903i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3420587921.342058792
L(12)L(\frac12) ≈\approx 1.3420587921.342058792
L(1)L(1) 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+(−0.634−0.773i)T 1 + (-0.634 - 0.773i)T
3 1+(0.956−0.290i)T 1 + (0.956 - 0.290i)T
5 1+(−0.995+0.0980i)T 1 + (-0.995 + 0.0980i)T
good7 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
11 1+(0.555−0.831i)T2 1 + (0.555 - 0.831i)T^{2}
13 1+(0.980+0.195i)T2 1 + (0.980 + 0.195i)T^{2}
17 1+(0.181+0.0750i)T+(0.707+0.707i)T2 1 + (0.181 + 0.0750i)T + (0.707 + 0.707i)T^{2}
19 1+(−1.11−1.36i)T+(−0.195+0.980i)T2 1 + (-1.11 - 1.36i)T + (-0.195 + 0.980i)T^{2}
23 1+(0.301−1.51i)T+(−0.923−0.382i)T2 1 + (0.301 - 1.51i)T + (-0.923 - 0.382i)T^{2}
29 1+(−0.555−0.831i)T2 1 + (-0.555 - 0.831i)T^{2}
31 1+(−0.275+0.275i)T−iT2 1 + (-0.275 + 0.275i)T - iT^{2}
37 1+(−0.195−0.980i)T2 1 + (-0.195 - 0.980i)T^{2}
41 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
43 1+(−0.831−0.555i)T2 1 + (-0.831 - 0.555i)T^{2}
47 1+(−0.222+0.536i)T+(−0.707−0.707i)T2 1 + (-0.222 + 0.536i)T + (-0.707 - 0.707i)T^{2}
53 1+(0.979−0.523i)T+(0.555−0.831i)T2 1 + (0.979 - 0.523i)T + (0.555 - 0.831i)T^{2}
59 1+(0.980−0.195i)T2 1 + (0.980 - 0.195i)T^{2}
61 1+(−0.273−0.902i)T+(−0.831+0.555i)T2 1 + (-0.273 - 0.902i)T + (-0.831 + 0.555i)T^{2}
67 1+(0.831−0.555i)T2 1 + (0.831 - 0.555i)T^{2}
71 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
73 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
79 1+(0.707+1.70i)T+(−0.707+0.707i)T2 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}
83 1+(1.28−1.05i)T+(0.195−0.980i)T2 1 + (1.28 - 1.05i)T + (0.195 - 0.980i)T^{2}
89 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
97 1+iT2 1 + iT^{2}
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   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

−9.710063343198948032123117765714, −8.914458830684251264390383865872, −7.80611658486697960200080713862, −7.08738827077406799728514403914, −6.18818644940989492691636294319, −5.64740147579137892277136501945, −5.13111038920741816657284227419, −4.11281964544669360056335574104, −3.18506696337845341771873356682, −1.62379898251794784201869436310, 0.984643075519604886101241121566, 2.13259106406426040867456880306, 3.02953465010534006398435480071, 4.50519715114672908358362805117, 5.00087583960738030352609225256, 5.87382609962848777705990551935, 6.48400335596443868232104254518, 7.18307688617482357398860830524, 8.577856217622875057706214584250, 9.548371537106066114425770405501

Graph of the ZZ-function along the critical line