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Properties

Label 2-3840-40.29-c1-0-47
Degree 22
Conductor 38403840
Sign 0.948+0.316i0.948 + 0.316i
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (−1 − 2i)5-s + 2i·7-s + 9-s + 2i·11-s − 2·13-s + (−1 − 2i)15-s − 6i·17-s + 8i·19-s + 2i·21-s − 4i·23-s + (−3 + 4i)25-s + 27-s − 8i·29-s + 2i·33-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.447 − 0.894i)5-s + 0.755i·7-s + 0.333·9-s + 0.603i·11-s − 0.554·13-s + (−0.258 − 0.516i)15-s − 1.45i·17-s + 1.83i·19-s + 0.436i·21-s − 0.834i·23-s + (−0.600 + 0.800i)25-s + 0.192·27-s − 1.48i·29-s + 0.348i·33-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.948+0.316i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=((0.948+0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.948+0.316i0.948 + 0.316i
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3840(2689,⋅)\chi_{3840} (2689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 0.948+0.316i)(2,\ 3840,\ (\ :1/2),\ 0.948 + 0.316i)

Particular Values

L(1)L(1) ≈\approx 2.0461571282.046157128
L(12)L(\frac12) ≈\approx 2.0461571282.046157128
L(32)L(\frac{3}{2}) 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 1
3 1−T 1 - T
5 1+(1+2i)T 1 + (1 + 2i)T
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1+2T+13T2 1 + 2T + 13T^{2}
17 1+6iT−17T2 1 + 6iT - 17T^{2}
19 1−8iT−19T2 1 - 8iT - 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1+8iT−29T2 1 + 8iT - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−10T+37T2 1 - 10T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−12T+43T2 1 - 12T + 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−10T+53T2 1 - 10T + 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1−2iT−61T2 1 - 2iT - 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1+4iT−73T2 1 + 4iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+8iT−97T2 1 + 8iT - 97T^{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

−8.454161731621803197736000328595, −7.73703881860251520331600841664, −7.32885717521294159283979550907, −6.09222664372060925189475261887, −5.43489347138969183726316710585, −4.49273717386838013180738857005, −4.02800632675067231706846133713, −2.74427696285614815105492608089, −2.10652803904730204309678777279, −0.76059464814911069789425674428, 0.846455618909862638156068365217, 2.26233135901830134180366625644, 3.07125601519601540624737078443, 3.81550548915274542901625141548, 4.48099107336196954487092116749, 5.60257831360239646988579279010, 6.54939893834362399380307207137, 7.17608752393453152373898427159, 7.66990402823539862264235295878, 8.472265957283386840669805876342

Graph of the ZZ-function along the critical line