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Properties

Label 2-3840-3840.1829-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(1829,⋅)\chi_{3840} (1829, \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 0.52011509550.5201150955
L(12)L(\frac12) ≈\approx 0.52011509550.5201150955
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

−8.943526750034332877729558779680, −8.432807935921744325454486527691, −7.44604976776645421650517867411, −6.27662854953858377005661018473, −5.33890657809636566502527653633, −4.86884408118812987694999941350, −4.05447520768519576978265086784, −3.56500832767071048576532429655, −2.30481221544252903400944467927, −1.07656549670854025249757332210, 0.37899552025106813163824562687, 2.07693893165569425493167294106, 3.07805373918776936566645187723, 4.32440576743959303554071273960, 4.91388820707285859750416352064, 6.01880985376400754085178704481, 6.51217020425965982867717597332, 6.98388091091048291964859618190, 7.66183636598370938439339919619, 8.345639401491680226357575679345

Graph of the ZZ-function along the critical line