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Properties

Label 2-200-40.29-c5-0-65
Degree 22
Conductor 200200
Sign −0.837+0.545i-0.837 + 0.545i
Analytic cond. 32.076732.0767
Root an. cond. 5.663635.66363
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.214 − 5.65i)2-s + 18.7·3-s + (−31.9 + 2.42i)4-s + (−4.03 − 106. i)6-s + 107. i·7-s + (20.5 + 179. i)8-s + 109.·9-s − 272. i·11-s + (−599. + 45.5i)12-s − 198.·13-s + (607. − 23.0i)14-s + (1.01e3 − 154. i)16-s − 2.06e3i·17-s + (−23.5 − 621. i)18-s − 1.89e3i·19-s + ⋯
L(s)  = 1  + (−0.0379 − 0.999i)2-s + 1.20·3-s + (−0.997 + 0.0757i)4-s + (−0.0457 − 1.20i)6-s + 0.829i·7-s + (0.113 + 0.993i)8-s + 0.452·9-s − 0.678i·11-s + (−1.20 + 0.0913i)12-s − 0.325·13-s + (0.828 − 0.0314i)14-s + (0.988 − 0.151i)16-s − 1.73i·17-s + (−0.0171 − 0.452i)18-s − 1.20i·19-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=((−0.837+0.545i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.545i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+5/2) L(s)=((−0.837+0.545i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: −0.837+0.545i-0.837 + 0.545i
Analytic conductor: 32.076732.0767
Root analytic conductor: 5.663635.66363
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ200(149,⋅)\chi_{200} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 200, ( :5/2), −0.837+0.545i)(2,\ 200,\ (\ :5/2),\ -0.837 + 0.545i)

Particular Values

L(3)L(3) ≈\approx 2.0001789832.000178983
L(12)L(\frac12) ≈\approx 2.0001789832.000178983
L(72)L(\frac{7}{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+(0.214+5.65i)T 1 + (0.214 + 5.65i)T
5 1 1
good3 1−18.7T+243T2 1 - 18.7T + 243T^{2}
7 1−107.iT−1.68e4T2 1 - 107. iT - 1.68e4T^{2}
11 1+272.iT−1.61e5T2 1 + 272. iT - 1.61e5T^{2}
13 1+198.T+3.71e5T2 1 + 198.T + 3.71e5T^{2}
17 1+2.06e3iT−1.41e6T2 1 + 2.06e3iT - 1.41e6T^{2}
19 1+1.89e3iT−2.47e6T2 1 + 1.89e3iT - 2.47e6T^{2}
23 1+987.iT−6.43e6T2 1 + 987. iT - 6.43e6T^{2}
29 1+8.01e3iT−2.05e7T2 1 + 8.01e3iT - 2.05e7T^{2}
31 1−827.T+2.86e7T2 1 - 827.T + 2.86e7T^{2}
37 1−9.42e3T+6.93e7T2 1 - 9.42e3T + 6.93e7T^{2}
41 1+8.22e3T+1.15e8T2 1 + 8.22e3T + 1.15e8T^{2}
43 1−9.30e3T+1.47e8T2 1 - 9.30e3T + 1.47e8T^{2}
47 1+1.38e4iT−2.29e8T2 1 + 1.38e4iT - 2.29e8T^{2}
53 1−2.77e4T+4.18e8T2 1 - 2.77e4T + 4.18e8T^{2}
59 1−2.51e4iT−7.14e8T2 1 - 2.51e4iT - 7.14e8T^{2}
61 1−2.64e4iT−8.44e8T2 1 - 2.64e4iT - 8.44e8T^{2}
67 1+3.85e4T+1.35e9T2 1 + 3.85e4T + 1.35e9T^{2}
71 1+7.10e4T+1.80e9T2 1 + 7.10e4T + 1.80e9T^{2}
73 1−1.86e4iT−2.07e9T2 1 - 1.86e4iT - 2.07e9T^{2}
79 1−7.55e4T+3.07e9T2 1 - 7.55e4T + 3.07e9T^{2}
83 1+1.25e5T+3.93e9T2 1 + 1.25e5T + 3.93e9T^{2}
89 1+3.03e4T+5.58e9T2 1 + 3.03e4T + 5.58e9T^{2}
97 1+1.56e4iT−8.58e9T2 1 + 1.56e4iT - 8.58e9T^{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

−11.41741951323517209949600270814, −9.991155544913944979439651214591, −9.112161537874882622053406471496, −8.626350946465473828877933587608, −7.47697667261892096017800574743, −5.61685796608592472252090006049, −4.32279561047950154368772725521, −2.81187431058671120894161946817, −2.48318532069844863634194154227, −0.53406277465308975867335877180, 1.55134694009352904798007621802, 3.46121859857541637989201904099, 4.34696826202529040576947244123, 5.87544460326345948155424841355, 7.17015136861881021997026954856, 7.910781529723529163151766850614, 8.718496984805164292501145731676, 9.757278627514257709056053828517, 10.55887415077192994466410158691, 12.45846450131474936915348055348

Graph of the ZZ-function along the critical line