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Properties

Label 2-200-40.29-c5-0-32
Degree 22
Conductor 200200
Sign −0.403−0.914i-0.403 - 0.914i
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  + (−3.87 + 4.12i)2-s + 18.2·3-s + (−2.00 − 31.9i)4-s + (−70.8 + 75.3i)6-s + 2.25i·7-s + (139. + 115. i)8-s + 91.3·9-s + 419. i·11-s + (−36.6 − 583. i)12-s + 106.·13-s + (−9.30 − 8.73i)14-s + (−1.01e3 + 127. i)16-s − 849. i·17-s + (−353. + 376. i)18-s + 335. i·19-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)2-s + 1.17·3-s + (−0.0625 − 0.998i)4-s + (−0.803 + 0.854i)6-s + 0.0173i·7-s + (0.770 + 0.637i)8-s + 0.375·9-s + 1.04i·11-s + (−0.0734 − 1.17i)12-s + 0.175·13-s + (−0.0126 − 0.0119i)14-s + (−0.992 + 0.124i)16-s − 0.712i·17-s + (−0.257 + 0.273i)18-s + 0.213i·19-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=((−0.403−0.914i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+5/2) L(s)=((−0.403−0.914i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: −0.403−0.914i-0.403 - 0.914i
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), −0.403−0.914i)(2,\ 200,\ (\ :5/2),\ -0.403 - 0.914i)

Particular Values

L(3)L(3) ≈\approx 1.7489368531.748936853
L(12)L(\frac12) ≈\approx 1.7489368531.748936853
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+(3.87−4.12i)T 1 + (3.87 - 4.12i)T
5 1 1
good3 1−18.2T+243T2 1 - 18.2T + 243T^{2}
7 1−2.25iT−1.68e4T2 1 - 2.25iT - 1.68e4T^{2}
11 1−419.iT−1.61e5T2 1 - 419. iT - 1.61e5T^{2}
13 1−106.T+3.71e5T2 1 - 106.T + 3.71e5T^{2}
17 1+849.iT−1.41e6T2 1 + 849. iT - 1.41e6T^{2}
19 1−335.iT−2.47e6T2 1 - 335. iT - 2.47e6T^{2}
23 1−3.54e3iT−6.43e6T2 1 - 3.54e3iT - 6.43e6T^{2}
29 1−5.20e3iT−2.05e7T2 1 - 5.20e3iT - 2.05e7T^{2}
31 1−5.63e3T+2.86e7T2 1 - 5.63e3T + 2.86e7T^{2}
37 1+61.9T+6.93e7T2 1 + 61.9T + 6.93e7T^{2}
41 1−1.62e4T+1.15e8T2 1 - 1.62e4T + 1.15e8T^{2}
43 1−2.41e3T+1.47e8T2 1 - 2.41e3T + 1.47e8T^{2}
47 1−2.27e4iT−2.29e8T2 1 - 2.27e4iT - 2.29e8T^{2}
53 1+1.36e4T+4.18e8T2 1 + 1.36e4T + 4.18e8T^{2}
59 1−2.34e4iT−7.14e8T2 1 - 2.34e4iT - 7.14e8T^{2}
61 1−3.34e4iT−8.44e8T2 1 - 3.34e4iT - 8.44e8T^{2}
67 1+6.61e4T+1.35e9T2 1 + 6.61e4T + 1.35e9T^{2}
71 1+5.14e4T+1.80e9T2 1 + 5.14e4T + 1.80e9T^{2}
73 1−2.12e4iT−2.07e9T2 1 - 2.12e4iT - 2.07e9T^{2}
79 1+3.84e4T+3.07e9T2 1 + 3.84e4T + 3.07e9T^{2}
83 1−9.31e4T+3.93e9T2 1 - 9.31e4T + 3.93e9T^{2}
89 1−6.06e4T+5.58e9T2 1 - 6.06e4T + 5.58e9T^{2}
97 1+1.57e5iT−8.58e9T2 1 + 1.57e5iT - 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

−11.80623882411953050216714869229, −10.52444536941850391701623867454, −9.470229599496823795802439142566, −8.979899407503517679321127699996, −7.78748229865208752314679041548, −7.23268057688480453584252795483, −5.79161525250931331495957751071, −4.42827878190827558835473356704, −2.76167325539945230337783813830, −1.41110812247419322215631489183, 0.59521633487709350141530689228, 2.20059524333434645640328131994, 3.15026141554560566253275711032, 4.22477650312082011786470813835, 6.25276012582467654539530878303, 7.76293779531959761338710414003, 8.455313306648328450451045385518, 9.104031937598414937963639832760, 10.22150620321435482775091823815, 11.10037973258859169704634939916

Graph of the ZZ-function along the critical line