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Properties

Label 2-200-40.29-c5-0-62
Degree 22
Conductor 200200
Sign −0.0565+0.998i-0.0565 + 0.998i
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  + (−5.60 + 0.765i)2-s + 17.3·3-s + (30.8 − 8.57i)4-s + (−97.0 + 13.2i)6-s − 9.19i·7-s + (−166. + 71.6i)8-s + 56.8·9-s + 160. i·11-s + (533. − 148. i)12-s − 368.·13-s + (7.03 + 51.5i)14-s + (876. − 528. i)16-s − 1.26e3i·17-s + (−318. + 43.4i)18-s − 2.48e3i·19-s + ⋯
L(s)  = 1  + (−0.990 + 0.135i)2-s + 1.11·3-s + (0.963 − 0.268i)4-s + (−1.10 + 0.150i)6-s − 0.0708i·7-s + (−0.918 + 0.395i)8-s + 0.233·9-s + 0.399i·11-s + (1.07 − 0.297i)12-s − 0.604·13-s + (0.00958 + 0.0702i)14-s + (0.856 − 0.516i)16-s − 1.05i·17-s + (−0.231 + 0.0316i)18-s − 1.58i·19-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=((−0.0565+0.998i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0565 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+5/2) L(s)=((−0.0565+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0565 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: −0.0565+0.998i-0.0565 + 0.998i
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), −0.0565+0.998i)(2,\ 200,\ (\ :5/2),\ -0.0565 + 0.998i)

Particular Values

L(3)L(3) ≈\approx 1.2259157881.225915788
L(12)L(\frac12) ≈\approx 1.2259157881.225915788
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+(5.60−0.765i)T 1 + (5.60 - 0.765i)T
5 1 1
good3 1−17.3T+243T2 1 - 17.3T + 243T^{2}
7 1+9.19iT−1.68e4T2 1 + 9.19iT - 1.68e4T^{2}
11 1−160.iT−1.61e5T2 1 - 160. iT - 1.61e5T^{2}
13 1+368.T+3.71e5T2 1 + 368.T + 3.71e5T^{2}
17 1+1.26e3iT−1.41e6T2 1 + 1.26e3iT - 1.41e6T^{2}
19 1+2.48e3iT−2.47e6T2 1 + 2.48e3iT - 2.47e6T^{2}
23 1−422.iT−6.43e6T2 1 - 422. iT - 6.43e6T^{2}
29 1+5.66e3iT−2.05e7T2 1 + 5.66e3iT - 2.05e7T^{2}
31 1−9.38e3T+2.86e7T2 1 - 9.38e3T + 2.86e7T^{2}
37 1+3.56e3T+6.93e7T2 1 + 3.56e3T + 6.93e7T^{2}
41 1+5.94e3T+1.15e8T2 1 + 5.94e3T + 1.15e8T^{2}
43 1+1.06e4T+1.47e8T2 1 + 1.06e4T + 1.47e8T^{2}
47 1−9.24e3iT−2.29e8T2 1 - 9.24e3iT - 2.29e8T^{2}
53 1+8.97e3T+4.18e8T2 1 + 8.97e3T + 4.18e8T^{2}
59 1+2.74e4iT−7.14e8T2 1 + 2.74e4iT - 7.14e8T^{2}
61 1+5.05e4iT−8.44e8T2 1 + 5.05e4iT - 8.44e8T^{2}
67 1+5.96e3T+1.35e9T2 1 + 5.96e3T + 1.35e9T^{2}
71 1−6.72e4T+1.80e9T2 1 - 6.72e4T + 1.80e9T^{2}
73 1+8.57e4iT−2.07e9T2 1 + 8.57e4iT - 2.07e9T^{2}
79 1+5.65e4T+3.07e9T2 1 + 5.65e4T + 3.07e9T^{2}
83 1−3.02e4T+3.93e9T2 1 - 3.02e4T + 3.93e9T^{2}
89 1+1.13e5T+5.58e9T2 1 + 1.13e5T + 5.58e9T^{2}
97 1+1.38e5iT−8.58e9T2 1 + 1.38e5iT - 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

−11.21674490758310487239033165469, −9.871762792024778912064414171280, −9.348925237016177579488179675499, −8.360697776758855776852228950804, −7.52836209191405280782951975617, −6.57868797251436501497605356574, −4.89300183238820448491923779379, −3.02627148606664269833690788294, −2.19104059147723937472580927974, −0.43032889807148529660421991648, 1.46682099910730364225002438531, 2.68415318251682251563530448994, 3.74114700066589416693045636734, 5.80917321002664058443489174106, 7.08864363342897289036343964055, 8.300982659859572060721508250131, 8.539249639978777502820053310504, 9.819217764058962127134790954003, 10.47487686935876802408947811017, 11.77947578501420200223326241220

Graph of the ZZ-function along the critical line