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Properties

Label 2-4788-133.132-c1-0-1
Degree 22
Conductor 47884788
Sign 0.0791+0.996i0.0791 + 0.996i
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  + 3.27i·5-s + (−2.63 + 0.209i)7-s − 6.50·11-s + 7.27i·17-s + 4.35i·19-s − 8.71·23-s − 5.72·25-s + (−0.685 − 8.63i)35-s + 11.8·43-s − 2.72i·47-s + (6.91 − 1.10i)49-s − 21.3i·55-s − 10.8i·61-s + 16.0i·73-s + (17.1 − 1.36i)77-s + ⋯
L(s)  = 1  + 1.46i·5-s + (−0.996 + 0.0791i)7-s − 1.96·11-s + 1.76i·17-s + 0.999i·19-s − 1.81·23-s − 1.14·25-s + (−0.115 − 1.45i)35-s + 1.80·43-s − 0.397i·47-s + (0.987 − 0.157i)49-s − 2.87i·55-s − 1.38i·61-s + 1.88i·73-s + (1.95 − 0.155i)77-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=((0.0791+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0791 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=((0.0791+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0791 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: 0.0791+0.996i0.0791 + 0.996i
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4788(3457,⋅)\chi_{4788} (3457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), 0.0791+0.996i)(2,\ 4788,\ (\ :1/2),\ 0.0791 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.11187940540.1118794054
L(12)L(\frac12) ≈\approx 0.11187940540.1118794054
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 1
7 1+(2.63−0.209i)T 1 + (2.63 - 0.209i)T
19 1−4.35iT 1 - 4.35iT
good5 1−3.27iT−5T2 1 - 3.27iT - 5T^{2}
11 1+6.50T+11T2 1 + 6.50T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−7.27iT−17T2 1 - 7.27iT - 17T^{2}
23 1+8.71T+23T2 1 + 8.71T + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−11.8T+43T2 1 - 11.8T + 43T^{2}
47 1+2.72iT−47T2 1 + 2.72iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+10.8iT−61T2 1 + 10.8iT - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1−16.0iT−73T2 1 - 16.0iT - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+16iT−83T2 1 + 16iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+97T2 1 + 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

−8.684191884764704245818278166236, −7.84815702202853388186755460797, −7.57702819285540442094873779143, −6.50015327742270569479729818810, −6.04239257736752281111941002112, −5.49343565777563395542004367468, −4.07122095561051138460243903168, −3.50750174785511774819188366317, −2.64962642282764680250501931564, −2.03218476892242346660688628487, 0.04084124142431136516061048982, 0.73675386368232727167534285122, 2.33236319269032641597618614584, 2.90818876092883664288765609777, 4.14053167441187102198583230992, 4.84893659703139497911491424112, 5.43377444597293221657423672040, 6.08879415266355529837215314443, 7.24069595421292759340134257382, 7.70537118235592702767235746238

Graph of the ZZ-function along the critical line