Properties

Label 2-200-40.29-c5-0-59
Degree 22
Conductor 200200
Sign 0.4030.915i0.403 - 0.915i
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  + (4.94 + 2.74i)2-s + 23.5·3-s + (16.8 + 27.1i)4-s + (116. + 64.7i)6-s + 39.5i·7-s + (8.81 + 180. i)8-s + 311.·9-s − 236. i·11-s + (397. + 639. i)12-s + 942.·13-s + (−108. + 195. i)14-s + (−453. + 918. i)16-s − 1.10e3i·17-s + (1.53e3 + 856. i)18-s + 2.31e3i·19-s + ⋯
L(s)  = 1  + (0.874 + 0.485i)2-s + 1.51·3-s + (0.527 + 0.849i)4-s + (1.32 + 0.733i)6-s + 0.305i·7-s + (0.0486 + 0.998i)8-s + 1.28·9-s − 0.589i·11-s + (0.797 + 1.28i)12-s + 1.54·13-s + (−0.148 + 0.266i)14-s + (−0.442 + 0.896i)16-s − 0.925i·17-s + (1.12 + 0.622i)18-s + 1.47i·19-s + ⋯

Functional equation

Λ(s)=(200s/2ΓC(s)L(s)=((0.4030.915i)Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2ΓC(s+5/2)L(s)=((0.4030.915i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23522^{3} \cdot 5^{2}
Sign: 0.4030.915i0.403 - 0.915i
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), 0.4030.915i)(2,\ 200,\ (\ :5/2),\ 0.403 - 0.915i)

Particular Values

L(3)L(3) \approx 5.9740877535.974087753
L(12)L(\frac12) \approx 5.9740877535.974087753
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(4.942.74i)T 1 + (-4.94 - 2.74i)T
5 1 1
good3 123.5T+243T2 1 - 23.5T + 243T^{2}
7 139.5iT1.68e4T2 1 - 39.5iT - 1.68e4T^{2}
11 1+236.iT1.61e5T2 1 + 236. iT - 1.61e5T^{2}
13 1942.T+3.71e5T2 1 - 942.T + 3.71e5T^{2}
17 1+1.10e3iT1.41e6T2 1 + 1.10e3iT - 1.41e6T^{2}
19 12.31e3iT2.47e6T2 1 - 2.31e3iT - 2.47e6T^{2}
23 1861.iT6.43e6T2 1 - 861. iT - 6.43e6T^{2}
29 11.66e3iT2.05e7T2 1 - 1.66e3iT - 2.05e7T^{2}
31 1+4.18e3T+2.86e7T2 1 + 4.18e3T + 2.86e7T^{2}
37 11.45e4T+6.93e7T2 1 - 1.45e4T + 6.93e7T^{2}
41 1+2.00e4T+1.15e8T2 1 + 2.00e4T + 1.15e8T^{2}
43 1+5.24e3T+1.47e8T2 1 + 5.24e3T + 1.47e8T^{2}
47 1+2.16e4iT2.29e8T2 1 + 2.16e4iT - 2.29e8T^{2}
53 1+1.49e4T+4.18e8T2 1 + 1.49e4T + 4.18e8T^{2}
59 1+4.24e4iT7.14e8T2 1 + 4.24e4iT - 7.14e8T^{2}
61 1+3.11e4iT8.44e8T2 1 + 3.11e4iT - 8.44e8T^{2}
67 1+2.78e4T+1.35e9T2 1 + 2.78e4T + 1.35e9T^{2}
71 14.23e4T+1.80e9T2 1 - 4.23e4T + 1.80e9T^{2}
73 1+1.17e4iT2.07e9T2 1 + 1.17e4iT - 2.07e9T^{2}
79 16.98e4T+3.07e9T2 1 - 6.98e4T + 3.07e9T^{2}
83 1+6.45e3T+3.93e9T2 1 + 6.45e3T + 3.93e9T^{2}
89 1+2.56e4T+5.58e9T2 1 + 2.56e4T + 5.58e9T^{2}
97 11.51e5iT8.58e9T2 1 - 1.51e5iT - 8.58e9T^{2}
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   L(s)=p j=12(1αj,pps)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

−12.01111849326713233366252608296, −10.95287543930087956953689542476, −9.433711677132999338213710325691, −8.426986814726835929080934231098, −7.919343978201733881988276299687, −6.58756314737593498305679988668, −5.43196443524764680544740084551, −3.79536470089855203791602441131, −3.21636302404407496970634896658, −1.82940664296443008607828149098, 1.32122389604748337673496048353, 2.52550865256725532870674391186, 3.60433716391253172123757477085, 4.47041793696328614695262569853, 6.14239880411160770018495813406, 7.29936796933131473106335123276, 8.510549476037181597073523464986, 9.412765549620005320602485278627, 10.49654078174791012693643698352, 11.39991002522366641352963003020

Graph of the ZZ-function along the critical line