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Properties

Label 2-1920-1920.749-c0-0-1
Degree 22
Conductor 19201920
Sign 0.336−0.941i0.336 - 0.941i
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  + (−0.0980 + 0.995i)2-s + (0.881 + 0.471i)3-s + (−0.980 − 0.195i)4-s + (0.634 − 0.773i)5-s + (−0.555 + 0.831i)6-s + (0.290 − 0.956i)8-s + (0.555 + 0.831i)9-s + (0.707 + 0.707i)10-s + (−0.773 − 0.634i)12-s + (0.923 − 0.382i)15-s + (0.923 + 0.382i)16-s + (1.42 + 0.591i)17-s + (−0.881 + 0.471i)18-s + (0.0569 − 0.577i)19-s + (−0.773 + 0.634i)20-s + ⋯
L(s)  = 1  + (−0.0980 + 0.995i)2-s + (0.881 + 0.471i)3-s + (−0.980 − 0.195i)4-s + (0.634 − 0.773i)5-s + (−0.555 + 0.831i)6-s + (0.290 − 0.956i)8-s + (0.555 + 0.831i)9-s + (0.707 + 0.707i)10-s + (−0.773 − 0.634i)12-s + (0.923 − 0.382i)15-s + (0.923 + 0.382i)16-s + (1.42 + 0.591i)17-s + (−0.881 + 0.471i)18-s + (0.0569 − 0.577i)19-s + (−0.773 + 0.634i)20-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((0.336−0.941i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s) L(s)=((0.336−0.941i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 0.336−0.941i0.336 - 0.941i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1920(749,⋅)\chi_{1920} (749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :0), 0.336−0.941i)(2,\ 1920,\ (\ :0),\ 0.336 - 0.941i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5341538251.534153825
L(12)L(\frac12) ≈\approx 1.5341538251.534153825
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+(0.0980−0.995i)T 1 + (0.0980 - 0.995i)T
3 1+(−0.881−0.471i)T 1 + (-0.881 - 0.471i)T
5 1+(−0.634+0.773i)T 1 + (-0.634 + 0.773i)T
good7 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
11 1+(−0.831−0.555i)T2 1 + (-0.831 - 0.555i)T^{2}
13 1+(−0.195+0.980i)T2 1 + (-0.195 + 0.980i)T^{2}
17 1+(−1.42−0.591i)T+(0.707+0.707i)T2 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2}
19 1+(−0.0569+0.577i)T+(−0.980−0.195i)T2 1 + (-0.0569 + 0.577i)T + (-0.980 - 0.195i)T^{2}
23 1+(1.95+0.388i)T+(0.923+0.382i)T2 1 + (1.95 + 0.388i)T + (0.923 + 0.382i)T^{2}
29 1+(0.831−0.555i)T2 1 + (0.831 - 0.555i)T^{2}
31 1+(−1.38+1.38i)T−iT2 1 + (-1.38 + 1.38i)T - iT^{2}
37 1+(−0.980+0.195i)T2 1 + (-0.980 + 0.195i)T^{2}
41 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
43 1+(−0.555+0.831i)T2 1 + (-0.555 + 0.831i)T^{2}
47 1+(0.360−0.871i)T+(−0.707−0.707i)T2 1 + (0.360 - 0.871i)T + (-0.707 - 0.707i)T^{2}
53 1+(0.482−1.59i)T+(−0.831−0.555i)T2 1 + (0.482 - 1.59i)T + (-0.831 - 0.555i)T^{2}
59 1+(−0.195−0.980i)T2 1 + (-0.195 - 0.980i)T^{2}
61 1+(0.902−1.68i)T+(−0.555−0.831i)T2 1 + (0.902 - 1.68i)T + (-0.555 - 0.831i)T^{2}
67 1+(0.555+0.831i)T2 1 + (0.555 + 0.831i)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+(1.10+0.108i)T+(0.980+0.195i)T2 1 + (1.10 + 0.108i)T + (0.980 + 0.195i)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

−9.432368360249981925513530863793, −8.673267742896213492070808892650, −7.999172403049501871719324969754, −7.54637381408742789366181124709, −6.13678851011703757313682903722, −5.73827498125706631555425105777, −4.55493480367973433229476209756, −4.15234568086816841321835889626, −2.80814472448383068038664529769, −1.41310141165737786345959094492, 1.42449204432030177266256087033, 2.27584919086735991163549378568, 3.22035348753197686404053804461, 3.75290717227223949467603968902, 5.11282173526076216748669919564, 6.06628392049662501716518513920, 7.03306124024960230859373740602, 7.969510256386482598574130056170, 8.407055011474932138125602031238, 9.613837840037194537662451454947

Graph of the ZZ-function along the critical line