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Properties

Label 2-3840-3840.1709-c0-0-0
Degree 22
Conductor 38403840
Sign 0.817+0.575i0.817 + 0.575i
Analytic cond. 1.916401.91640
Root an. cond. 1.384341.38434
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  + (0.0490 − 0.998i)2-s + (0.514 + 0.857i)3-s + (−0.995 − 0.0980i)4-s + (0.336 − 0.941i)5-s + (0.881 − 0.471i)6-s + (−0.146 + 0.989i)8-s + (−0.471 + 0.881i)9-s + (−0.923 − 0.382i)10-s + (−0.427 − 0.903i)12-s + (0.980 − 0.195i)15-s + (0.980 + 0.195i)16-s + (1.77 + 0.352i)17-s + (0.857 + 0.514i)18-s + (0.293 + 0.0143i)19-s + (−0.427 + 0.903i)20-s + ⋯
L(s)  = 1  + (0.0490 − 0.998i)2-s + (0.514 + 0.857i)3-s + (−0.995 − 0.0980i)4-s + (0.336 − 0.941i)5-s + (0.881 − 0.471i)6-s + (−0.146 + 0.989i)8-s + (−0.471 + 0.881i)9-s + (−0.923 − 0.382i)10-s + (−0.427 − 0.903i)12-s + (0.980 − 0.195i)15-s + (0.980 + 0.195i)16-s + (1.77 + 0.352i)17-s + (0.857 + 0.514i)18-s + (0.293 + 0.0143i)19-s + (−0.427 + 0.903i)20-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.817+0.575i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s) L(s)=((0.817+0.575i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.817+0.575i0.817 + 0.575i
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3840(1709,⋅)\chi_{3840} (1709, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :0), 0.817+0.575i)(2,\ 3840,\ (\ :0),\ 0.817 + 0.575i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5477961711.547796171
L(12)L(\frac12) ≈\approx 1.5477961711.547796171
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+(−0.0490+0.998i)T 1 + (-0.0490 + 0.998i)T
3 1+(−0.514−0.857i)T 1 + (-0.514 - 0.857i)T
5 1+(−0.336+0.941i)T 1 + (-0.336 + 0.941i)T
good7 1+(−0.831−0.555i)T2 1 + (-0.831 - 0.555i)T^{2}
11 1+(−0.290+0.956i)T2 1 + (-0.290 + 0.956i)T^{2}
13 1+(0.634+0.773i)T2 1 + (0.634 + 0.773i)T^{2}
17 1+(−1.77−0.352i)T+(0.923+0.382i)T2 1 + (-1.77 - 0.352i)T + (0.923 + 0.382i)T^{2}
19 1+(−0.293−0.0143i)T+(0.995+0.0980i)T2 1 + (-0.293 - 0.0143i)T + (0.995 + 0.0980i)T^{2}
23 1+(0.145−1.47i)T+(−0.980−0.195i)T2 1 + (0.145 - 1.47i)T + (-0.980 - 0.195i)T^{2}
29 1+(−0.956+0.290i)T2 1 + (-0.956 + 0.290i)T^{2}
31 1+(−0.181+0.0750i)T+(0.707−0.707i)T2 1 + (-0.181 + 0.0750i)T + (0.707 - 0.707i)T^{2}
37 1+(−0.0980−0.995i)T2 1 + (-0.0980 - 0.995i)T^{2}
41 1+(−0.195+0.980i)T2 1 + (-0.195 + 0.980i)T^{2}
43 1+(−0.471−0.881i)T2 1 + (-0.471 - 0.881i)T^{2}
47 1+(−0.404+0.269i)T+(0.382−0.923i)T2 1 + (-0.404 + 0.269i)T + (0.382 - 0.923i)T^{2}
53 1+(−0.574−0.0851i)T+(0.956+0.290i)T2 1 + (-0.574 - 0.0851i)T + (0.956 + 0.290i)T^{2}
59 1+(0.634−0.773i)T2 1 + (0.634 - 0.773i)T^{2}
61 1+(−0.390+1.55i)T+(−0.881−0.471i)T2 1 + (-0.390 + 1.55i)T + (-0.881 - 0.471i)T^{2}
67 1+(−0.881−0.471i)T2 1 + (-0.881 - 0.471i)T^{2}
71 1+(−0.555+0.831i)T2 1 + (-0.555 + 0.831i)T^{2}
73 1+(−0.831+0.555i)T2 1 + (-0.831 + 0.555i)T^{2}
79 1+(−1.08+1.63i)T+(−0.382−0.923i)T2 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2}
83 1+(0.698−0.633i)T+(0.0980−0.995i)T2 1 + (0.698 - 0.633i)T + (0.0980 - 0.995i)T^{2}
89 1+(0.980−0.195i)T2 1 + (0.980 - 0.195i)T^{2}
97 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{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

−8.837703990590793528774063406132, −8.108713225245269429552442709807, −7.58861360629436449747737538757, −5.81213523759663097261622829203, −5.42538906509820984018495801499, −4.68437751640060550352855704386, −3.80029463018264226988793104407, −3.26770488388349878007206861585, −2.14631313322890035633347390395, −1.18269664870809742316742353317, 1.03271788724943338728817853611, 2.48119059993139024404462850774, 3.24519577524614751878416059524, 4.08986063915269881783256035517, 5.39449431734496505588286478940, 5.91113326906677415115268811953, 6.73010913645476971330694824505, 7.19030813674186631523910751942, 7.85475319896357741422668171968, 8.457126180747608084529700383251

Graph of the ZZ-function along the critical line