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Properties

Label 2-1881-209.208-c1-0-61
Degree 22
Conductor 18811881
Sign 0.893+0.448i0.893 + 0.448i
Analytic cond. 15.019815.0198
Root an. cond. 3.875543.87554
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  + 1.15·2-s − 0.654·4-s + 1.65·5-s + 0.0488i·7-s − 3.07·8-s + 1.91·10-s + (−1.85 − 2.74i)11-s + 3.48·13-s + 0.0566i·14-s − 2.26·16-s + 4.74i·17-s + (3.79 − 2.13i)19-s − 1.08·20-s + (−2.15 − 3.18i)22-s + 9.20·23-s + ⋯
L(s)  = 1  + 0.820·2-s − 0.327·4-s + 0.740·5-s + 0.0184i·7-s − 1.08·8-s + 0.606·10-s + (−0.559 − 0.828i)11-s + 0.966·13-s + 0.0151i·14-s − 0.565·16-s + 1.15i·17-s + (0.871 − 0.489i)19-s − 0.242·20-s + (−0.458 − 0.679i)22-s + 1.91·23-s + ⋯

Functional equation

Λ(s)=(1881s/2 ΓC(s) L(s)=((0.893+0.448i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1881s/2 ΓC(s+1/2) L(s)=((0.893+0.448i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18811881    =    32⋅11⋅193^{2} \cdot 11 \cdot 19
Sign: 0.893+0.448i0.893 + 0.448i
Analytic conductor: 15.019815.0198
Root analytic conductor: 3.875543.87554
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1881(208,⋅)\chi_{1881} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1881, ( :1/2), 0.893+0.448i)(2,\ 1881,\ (\ :1/2),\ 0.893 + 0.448i)

Particular Values

L(1)L(1) ≈\approx 2.5630273132.563027313
L(12)L(\frac12) ≈\approx 2.5630273132.563027313
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)
bad3 1 1
11 1+(1.85+2.74i)T 1 + (1.85 + 2.74i)T
19 1+(−3.79+2.13i)T 1 + (-3.79 + 2.13i)T
good2 1−1.15T+2T2 1 - 1.15T + 2T^{2}
5 1−1.65T+5T2 1 - 1.65T + 5T^{2}
7 1−0.0488iT−7T2 1 - 0.0488iT - 7T^{2}
13 1−3.48T+13T2 1 - 3.48T + 13T^{2}
17 1−4.74iT−17T2 1 - 4.74iT - 17T^{2}
23 1−9.20T+23T2 1 - 9.20T + 23T^{2}
29 1+1.20T+29T2 1 + 1.20T + 29T^{2}
31 1+10.0iT−31T2 1 + 10.0iT - 31T^{2}
37 1+4.13iT−37T2 1 + 4.13iT - 37T^{2}
41 1−8.22T+41T2 1 - 8.22T + 41T^{2}
43 1−2.11iT−43T2 1 - 2.11iT - 43T^{2}
47 1−1.20T+47T2 1 - 1.20T + 47T^{2}
53 1−0.192iT−53T2 1 - 0.192iT - 53T^{2}
59 1−0.879iT−59T2 1 - 0.879iT - 59T^{2}
61 1+10.7iT−61T2 1 + 10.7iT - 61T^{2}
67 1+5.05iT−67T2 1 + 5.05iT - 67T^{2}
71 1−0.916iT−71T2 1 - 0.916iT - 71T^{2}
73 1−5.88iT−73T2 1 - 5.88iT - 73T^{2}
79 1−1.29T+79T2 1 - 1.29T + 79T^{2}
83 1−5.12iT−83T2 1 - 5.12iT - 83T^{2}
89 1−13.1iT−89T2 1 - 13.1iT - 89T^{2}
97 1+9.01iT−97T2 1 + 9.01iT - 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

−9.186903664641348295929806400177, −8.507431732969086424363816636014, −7.61439338056828284810006078422, −6.38553834370371397286084293210, −5.77801839043902936051159373787, −5.29965625536582735685026580573, −4.17347593530681433461762321034, −3.39635037526339095134303736132, −2.44900352085186138322639539987, −0.880992957738986293661529412816, 1.16711426054461409055723946331, 2.64430738188761869250258511138, 3.39443758811537234228988102202, 4.53761112319491436324291856602, 5.23292481910070671989979385041, 5.77152636507264423280931393879, 6.80641120326945709632520733992, 7.53726220779755925712982135661, 8.758656130370347734630803040661, 9.220937538730009615748864219532

Graph of the ZZ-function along the critical line