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Properties

Label 2-3872-8.5-c1-0-98
Degree 22
Conductor 38723872
Sign 0.243−0.969i0.243 - 0.969i
Analytic cond. 30.918030.9180
Root an. cond. 5.560405.56040
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  − 2.85i·3-s − 2.60i·5-s + 0.196·7-s − 5.16·9-s + 3.97i·13-s − 7.45·15-s + 1.23·17-s − 5.47i·19-s − 0.560i·21-s − 3.19·23-s − 1.80·25-s + 6.17i·27-s + 3.03i·29-s − 9.83·31-s − 0.512i·35-s + ⋯
L(s)  = 1  − 1.64i·3-s − 1.16i·5-s + 0.0742·7-s − 1.72·9-s + 1.10i·13-s − 1.92·15-s + 0.300·17-s − 1.25i·19-s − 0.122i·21-s − 0.666·23-s − 0.360·25-s + 1.18i·27-s + 0.563i·29-s − 1.76·31-s − 0.0865i·35-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((0.243−0.969i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s+1/2) L(s)=((0.243−0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 0.243−0.969i0.243 - 0.969i
Analytic conductor: 30.918030.9180
Root analytic conductor: 5.560405.56040
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3872(1937,⋅)\chi_{3872} (1937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :1/2), 0.243−0.969i)(2,\ 3872,\ (\ :1/2),\ 0.243 - 0.969i)

Particular Values

L(1)L(1) ≈\approx 0.29282581590.2928258159
L(12)L(\frac12) ≈\approx 0.29282581590.2928258159
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
11 1 1
good3 1+2.85iT−3T2 1 + 2.85iT - 3T^{2}
5 1+2.60iT−5T2 1 + 2.60iT - 5T^{2}
7 1−0.196T+7T2 1 - 0.196T + 7T^{2}
13 1−3.97iT−13T2 1 - 3.97iT - 13T^{2}
17 1−1.23T+17T2 1 - 1.23T + 17T^{2}
19 1+5.47iT−19T2 1 + 5.47iT - 19T^{2}
23 1+3.19T+23T2 1 + 3.19T + 23T^{2}
29 1−3.03iT−29T2 1 - 3.03iT - 29T^{2}
31 1+9.83T+31T2 1 + 9.83T + 31T^{2}
37 1−7.55iT−37T2 1 - 7.55iT - 37T^{2}
41 1+7.31T+41T2 1 + 7.31T + 41T^{2}
43 1+3.82iT−43T2 1 + 3.82iT - 43T^{2}
47 1−0.507T+47T2 1 - 0.507T + 47T^{2}
53 1+0.0213iT−53T2 1 + 0.0213iT - 53T^{2}
59 1−5.64iT−59T2 1 - 5.64iT - 59T^{2}
61 1−12.9iT−61T2 1 - 12.9iT - 61T^{2}
67 1−2.16iT−67T2 1 - 2.16iT - 67T^{2}
71 1+4.27T+71T2 1 + 4.27T + 71T^{2}
73 1−2.56T+73T2 1 - 2.56T + 73T^{2}
79 1+5.10T+79T2 1 + 5.10T + 79T^{2}
83 1+2.30iT−83T2 1 + 2.30iT - 83T^{2}
89 1+6.22T+89T2 1 + 6.22T + 89T^{2}
97 1+0.832T+97T2 1 + 0.832T + 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

−7.85295501670039422854855109277, −7.08670083136240732446836257603, −6.67917004533262325441754559304, −5.70976721869674262319191711194, −5.03446912420936518980417855427, −4.14507277262024451274887054862, −2.89226187924356117245016918531, −1.81522674745073099004026790570, −1.29088061501421870988151453388, −0.082562527925508140720580206290, 2.03519449112753642198895404299, 3.28989818844095098577479407691, 3.46323233814996217893541986383, 4.39051423560053834181576515485, 5.42949252951551065069383224617, 5.80002031214632792367038400613, 6.78955715113315561051978476650, 7.79212205455881247444257322674, 8.273155251224223254086702966536, 9.364864773571584933135748579059

Graph of the ZZ-function along the critical line