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Properties

Label 2-1682-29.28-c1-0-53
Degree 22
Conductor 16821682
Sign 0.233+0.972i0.233 + 0.972i
Analytic cond. 13.430813.4308
Root an. cond. 3.664813.66481
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  − i·2-s − 0.710i·3-s − 4-s + 3.33·5-s − 0.710·6-s + 4.40·7-s + i·8-s + 2.49·9-s − 3.33i·10-s + 2.39i·11-s + 0.710i·12-s − 5.40·13-s − 4.40i·14-s − 2.36i·15-s + 16-s − 3.10i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.409i·3-s − 0.5·4-s + 1.49·5-s − 0.289·6-s + 1.66·7-s + 0.353i·8-s + 0.831·9-s − 1.05i·10-s + 0.723i·11-s + 0.204i·12-s − 1.49·13-s − 1.17i·14-s − 0.611i·15-s + 0.250·16-s − 0.753i·17-s + ⋯

Functional equation

Λ(s)=(1682s/2 ΓC(s) L(s)=((0.233+0.972i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1682s/2 ΓC(s+1/2) L(s)=((0.233+0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16821682    =    2⋅2922 \cdot 29^{2}
Sign: 0.233+0.972i0.233 + 0.972i
Analytic conductor: 13.430813.4308
Root analytic conductor: 3.664813.66481
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1682(1681,⋅)\chi_{1682} (1681, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1682, ( :1/2), 0.233+0.972i)(2,\ 1682,\ (\ :1/2),\ 0.233 + 0.972i)

Particular Values

L(1)L(1) ≈\approx 2.6478317982.647831798
L(12)L(\frac12) ≈\approx 2.6478317982.647831798
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+iT 1 + iT
29 1 1
good3 1+0.710iT−3T2 1 + 0.710iT - 3T^{2}
5 1−3.33T+5T2 1 - 3.33T + 5T^{2}
7 1−4.40T+7T2 1 - 4.40T + 7T^{2}
11 1−2.39iT−11T2 1 - 2.39iT - 11T^{2}
13 1+5.40T+13T2 1 + 5.40T + 13T^{2}
17 1+3.10iT−17T2 1 + 3.10iT - 17T^{2}
19 1+3.55iT−19T2 1 + 3.55iT - 19T^{2}
23 1+0.408T+23T2 1 + 0.408T + 23T^{2}
31 1−1.23iT−31T2 1 - 1.23iT - 31T^{2}
37 1−6.21iT−37T2 1 - 6.21iT - 37T^{2}
41 1+3.97iT−41T2 1 + 3.97iT - 41T^{2}
43 1+6.00iT−43T2 1 + 6.00iT - 43T^{2}
47 1−8.56iT−47T2 1 - 8.56iT - 47T^{2}
53 1−5.49T+53T2 1 - 5.49T + 53T^{2}
59 1−8.70T+59T2 1 - 8.70T + 59T^{2}
61 1−11.6iT−61T2 1 - 11.6iT - 61T^{2}
67 1+8.27T+67T2 1 + 8.27T + 67T^{2}
71 1+4.70T+71T2 1 + 4.70T + 71T^{2}
73 1+2.91iT−73T2 1 + 2.91iT - 73T^{2}
79 1+5.81iT−79T2 1 + 5.81iT - 79T^{2}
83 1+7.60T+83T2 1 + 7.60T + 83T^{2}
89 1−10.6iT−89T2 1 - 10.6iT - 89T^{2}
97 1+16.9iT−97T2 1 + 16.9iT - 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

−9.392242853942785505404988898178, −8.573876872574147112108976053249, −7.41441452469863887903655251969, −7.07145298898570914768697585827, −5.68990853561515007527602756686, −4.84318988149459090995843144720, −4.52315268520743071957567093297, −2.56756896218501661480904895545, −2.04417911003482456662543305314, −1.21131686720096201216714704877, 1.42320691757046135135956033633, 2.27627223719113904449528713559, 3.93889840920705536850731612216, 4.86837399489284619635976424497, 5.38181724819280566264068352464, 6.14363949678759785520239893663, 7.18518700405543176701410890370, 7.925132841357985671795056841615, 8.681714745600964449079617941820, 9.560845736556520019438282604609

Graph of the ZZ-function along the critical line