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Properties

Label 2-3840-3840.1469-c0-0-0
Degree 22
Conductor 38403840
Sign 0.359+0.932i0.359 + 0.932i
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  + (0.857 − 0.514i)2-s + (−0.941 + 0.336i)3-s + (0.471 − 0.881i)4-s + (0.803 − 0.595i)5-s + (−0.634 + 0.773i)6-s + (−0.0490 − 0.998i)8-s + (0.773 − 0.634i)9-s + (0.382 − 0.923i)10-s + (−0.146 + 0.989i)12-s + (−0.555 + 0.831i)15-s + (−0.555 − 0.831i)16-s + (1.09 + 1.64i)17-s + (0.336 − 0.941i)18-s + (0.0504 + 0.0841i)19-s + (−0.146 − 0.989i)20-s + ⋯
L(s)  = 1  + (0.857 − 0.514i)2-s + (−0.941 + 0.336i)3-s + (0.471 − 0.881i)4-s + (0.803 − 0.595i)5-s + (−0.634 + 0.773i)6-s + (−0.0490 − 0.998i)8-s + (0.773 − 0.634i)9-s + (0.382 − 0.923i)10-s + (−0.146 + 0.989i)12-s + (−0.555 + 0.831i)15-s + (−0.555 − 0.831i)16-s + (1.09 + 1.64i)17-s + (0.336 − 0.941i)18-s + (0.0504 + 0.0841i)19-s + (−0.146 − 0.989i)20-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.359+0.932i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s) L(s)=((0.359+0.932i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.359+0.932i0.359 + 0.932i
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3840(1469,⋅)\chi_{3840} (1469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :0), 0.359+0.932i)(2,\ 3840,\ (\ :0),\ 0.359 + 0.932i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9552988911.955298891
L(12)L(\frac12) ≈\approx 1.9552988911.955298891
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+(−0.857+0.514i)T 1 + (-0.857 + 0.514i)T
3 1+(0.941−0.336i)T 1 + (0.941 - 0.336i)T
5 1+(−0.803+0.595i)T 1 + (-0.803 + 0.595i)T
good7 1+(−0.980+0.195i)T2 1 + (-0.980 + 0.195i)T^{2}
11 1+(0.0980+0.995i)T2 1 + (0.0980 + 0.995i)T^{2}
13 1+(0.956−0.290i)T2 1 + (0.956 - 0.290i)T^{2}
17 1+(−1.09−1.64i)T+(−0.382+0.923i)T2 1 + (-1.09 - 1.64i)T + (-0.382 + 0.923i)T^{2}
19 1+(−0.0504−0.0841i)T+(−0.471+0.881i)T2 1 + (-0.0504 - 0.0841i)T + (-0.471 + 0.881i)T^{2}
23 1+(−1.71−0.914i)T+(0.555+0.831i)T2 1 + (-1.71 - 0.914i)T + (0.555 + 0.831i)T^{2}
29 1+(−0.995−0.0980i)T2 1 + (-0.995 - 0.0980i)T^{2}
31 1+(0.674+1.62i)T+(−0.707+0.707i)T2 1 + (0.674 + 1.62i)T + (-0.707 + 0.707i)T^{2}
37 1+(−0.881+0.471i)T2 1 + (-0.881 + 0.471i)T^{2}
41 1+(0.831−0.555i)T2 1 + (0.831 - 0.555i)T^{2}
43 1+(0.773+0.634i)T2 1 + (0.773 + 0.634i)T^{2}
47 1+(1.77+0.352i)T+(0.923+0.382i)T2 1 + (1.77 + 0.352i)T + (0.923 + 0.382i)T^{2}
53 1+(−0.195+0.00961i)T+(0.995−0.0980i)T2 1 + (-0.195 + 0.00961i)T + (0.995 - 0.0980i)T^{2}
59 1+(0.956+0.290i)T2 1 + (0.956 + 0.290i)T^{2}
61 1+(1.21+0.574i)T+(0.634+0.773i)T2 1 + (1.21 + 0.574i)T + (0.634 + 0.773i)T^{2}
67 1+(0.634+0.773i)T2 1 + (0.634 + 0.773i)T^{2}
71 1+(0.195+0.980i)T2 1 + (0.195 + 0.980i)T^{2}
73 1+(−0.980−0.195i)T2 1 + (-0.980 - 0.195i)T^{2}
79 1+(−0.216−1.08i)T+(−0.923+0.382i)T2 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2}
83 1+(1.49+0.375i)T+(0.881+0.471i)T2 1 + (1.49 + 0.375i)T + (0.881 + 0.471i)T^{2}
89 1+(−0.555+0.831i)T2 1 + (-0.555 + 0.831i)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

−8.754976334151284026313872981123, −7.59610945463091127382222511223, −6.68633134827692936170181548434, −5.89347611929800609325375905935, −5.56348808787034171734903330177, −4.85266427698434991760666515872, −4.01763698442994564838030174297, −3.24506776499063633577370946611, −1.85587927076272745812810865765, −1.11080994432832434746151460933, 1.38871044047477092385636699953, 2.67655413588398577722202642395, 3.26797546154278265673661624711, 4.70108375886782294619843086255, 5.14532108604631468462947809723, 5.77179392305812511165414418044, 6.61573008516313996191927616692, 7.07725875475548308791239990399, 7.56883017986990507577668825032, 8.731382549967848477030247734676

Graph of the ZZ-function along the critical line