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Properties

Label 2-1920-1920.509-c0-0-1
Degree 22
Conductor 19201920
Sign 0.427+0.903i0.427 + 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  + (−0.881 − 0.471i)2-s + (−0.634 − 0.773i)3-s + (0.555 + 0.831i)4-s + (0.956 + 0.290i)5-s + (0.195 + 0.980i)6-s + (−0.0980 − 0.995i)8-s + (−0.195 + 0.980i)9-s + (−0.707 − 0.707i)10-s + (0.290 − 0.956i)12-s + (−0.382 − 0.923i)15-s + (−0.382 + 0.923i)16-s + (0.222 − 0.536i)17-s + (0.634 − 0.773i)18-s + (−0.172 − 0.0924i)19-s + (0.290 + 0.956i)20-s + ⋯
L(s)  = 1  + (−0.881 − 0.471i)2-s + (−0.634 − 0.773i)3-s + (0.555 + 0.831i)4-s + (0.956 + 0.290i)5-s + (0.195 + 0.980i)6-s + (−0.0980 − 0.995i)8-s + (−0.195 + 0.980i)9-s + (−0.707 − 0.707i)10-s + (0.290 − 0.956i)12-s + (−0.382 − 0.923i)15-s + (−0.382 + 0.923i)16-s + (0.222 − 0.536i)17-s + (0.634 − 0.773i)18-s + (−0.172 − 0.0924i)19-s + (0.290 + 0.956i)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+0.903i0.427 + 0.903i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1920(509,⋅)\chi_{1920} (509, \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 0.73531931270.7353193127
L(12)L(\frac12) ≈\approx 0.73531931270.7353193127
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+(0.881+0.471i)T 1 + (0.881 + 0.471i)T
3 1+(0.634+0.773i)T 1 + (0.634 + 0.773i)T
5 1+(−0.956−0.290i)T 1 + (-0.956 - 0.290i)T
good7 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
11 1+(−0.980+0.195i)T2 1 + (-0.980 + 0.195i)T^{2}
13 1+(0.831−0.555i)T2 1 + (0.831 - 0.555i)T^{2}
17 1+(−0.222+0.536i)T+(−0.707−0.707i)T2 1 + (-0.222 + 0.536i)T + (-0.707 - 0.707i)T^{2}
19 1+(0.172+0.0924i)T+(0.555+0.831i)T2 1 + (0.172 + 0.0924i)T + (0.555 + 0.831i)T^{2}
23 1+(0.523+0.783i)T+(−0.382+0.923i)T2 1 + (0.523 + 0.783i)T + (-0.382 + 0.923i)T^{2}
29 1+(0.980+0.195i)T2 1 + (0.980 + 0.195i)T^{2}
31 1+(−0.785+0.785i)T−iT2 1 + (-0.785 + 0.785i)T - iT^{2}
37 1+(0.555−0.831i)T2 1 + (0.555 - 0.831i)T^{2}
41 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
43 1+(0.195+0.980i)T2 1 + (0.195 + 0.980i)T^{2}
47 1+(−1.42−0.591i)T+(0.707+0.707i)T2 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2}
53 1+(−0.192−1.95i)T+(−0.980+0.195i)T2 1 + (-0.192 - 1.95i)T + (-0.980 + 0.195i)T^{2}
59 1+(0.831+0.555i)T2 1 + (0.831 + 0.555i)T^{2}
61 1+(−1.53+1.26i)T+(0.195−0.980i)T2 1 + (-1.53 + 1.26i)T + (0.195 - 0.980i)T^{2}
67 1+(−0.195+0.980i)T2 1 + (-0.195 + 0.980i)T^{2}
71 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
73 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
79 1+(−0.707+0.292i)T+(0.707−0.707i)T2 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2}
83 1+(0.183−0.344i)T+(−0.555−0.831i)T2 1 + (0.183 - 0.344i)T + (-0.555 - 0.831i)T^{2}
89 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)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

−9.350264389525125664447554021256, −8.498332861024472321638217686866, −7.68305960502946068500961794327, −6.93853123448907911416570976471, −6.27642338335382965564845282758, −5.52274470801818732466002294383, −4.28772375767253605765976746639, −2.76740480763780898880604927769, −2.16144952567403029596387187040, −0.959661349615277954646785574665, 1.15354620035108706012451273534, 2.42175024666545382190412833534, 3.86566418969955950609449214069, 5.06907440870230373123526592844, 5.60456870474248319389705811869, 6.31042114450035100085794282855, 7.03761495089841367111905402833, 8.214654754040759234073335373485, 8.893409258278248382987935115297, 9.548385508063944669832165982296

Graph of the ZZ-function along the critical line