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Properties

Label 2-200-40.29-c5-0-55
Degree 22
Conductor 200200
Sign 0.877+0.478i0.877 + 0.478i
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  + (−3.75 − 4.22i)2-s + 25.0·3-s + (−3.78 + 31.7i)4-s + (−94.1 − 105. i)6-s − 103. i·7-s + (148. − 103. i)8-s + 384.·9-s + 740. i·11-s + (−94.7 + 796. i)12-s + 892.·13-s + (−438. + 389. i)14-s + (−995. − 240. i)16-s − 1.13e3i·17-s + (−1.44e3 − 1.62e3i)18-s + 1.15e3i·19-s + ⋯
L(s)  = 1  + (−0.664 − 0.747i)2-s + 1.60·3-s + (−0.118 + 0.992i)4-s + (−1.06 − 1.20i)6-s − 0.799i·7-s + (0.820 − 0.571i)8-s + 1.58·9-s + 1.84i·11-s + (−0.189 + 1.59i)12-s + 1.46·13-s + (−0.597 + 0.530i)14-s + (−0.972 − 0.234i)16-s − 0.955i·17-s + (−1.05 − 1.18i)18-s + 0.734i·19-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=((0.877+0.478i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.478i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+5/2) L(s)=((0.877+0.478i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: 0.877+0.478i0.877 + 0.478i
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), 0.877+0.478i)(2,\ 200,\ (\ :5/2),\ 0.877 + 0.478i)

Particular Values

L(3)L(3) ≈\approx 2.7334574332.733457433
L(12)L(\frac12) ≈\approx 2.7334574332.733457433
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+(3.75+4.22i)T 1 + (3.75 + 4.22i)T
5 1 1
good3 1−25.0T+243T2 1 - 25.0T + 243T^{2}
7 1+103.iT−1.68e4T2 1 + 103. iT - 1.68e4T^{2}
11 1−740.iT−1.61e5T2 1 - 740. iT - 1.61e5T^{2}
13 1−892.T+3.71e5T2 1 - 892.T + 3.71e5T^{2}
17 1+1.13e3iT−1.41e6T2 1 + 1.13e3iT - 1.41e6T^{2}
19 1−1.15e3iT−2.47e6T2 1 - 1.15e3iT - 2.47e6T^{2}
23 1−1.60e3iT−6.43e6T2 1 - 1.60e3iT - 6.43e6T^{2}
29 1+2.15e3iT−2.05e7T2 1 + 2.15e3iT - 2.05e7T^{2}
31 1−4.95e3T+2.86e7T2 1 - 4.95e3T + 2.86e7T^{2}
37 1−4.40e3T+6.93e7T2 1 - 4.40e3T + 6.93e7T^{2}
41 1−3.78e3T+1.15e8T2 1 - 3.78e3T + 1.15e8T^{2}
43 1+1.30e4T+1.47e8T2 1 + 1.30e4T + 1.47e8T^{2}
47 1−8.00e3iT−2.29e8T2 1 - 8.00e3iT - 2.29e8T^{2}
53 1−3.43e4T+4.18e8T2 1 - 3.43e4T + 4.18e8T^{2}
59 1+2.20e4iT−7.14e8T2 1 + 2.20e4iT - 7.14e8T^{2}
61 1−2.82e3iT−8.44e8T2 1 - 2.82e3iT - 8.44e8T^{2}
67 1−5.49e4T+1.35e9T2 1 - 5.49e4T + 1.35e9T^{2}
71 1−4.28e4T+1.80e9T2 1 - 4.28e4T + 1.80e9T^{2}
73 1+2.08e4iT−2.07e9T2 1 + 2.08e4iT - 2.07e9T^{2}
79 1+3.02e4T+3.07e9T2 1 + 3.02e4T + 3.07e9T^{2}
83 1+9.39e4T+3.93e9T2 1 + 9.39e4T + 3.93e9T^{2}
89 1+1.17e3T+5.58e9T2 1 + 1.17e3T + 5.58e9T^{2}
97 1+7.41e4iT−8.58e9T2 1 + 7.41e4iT - 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

−11.38507466963049082752801211809, −10.06040779764080395934479630238, −9.667672272185071130212273909565, −8.578294608770426196125314651624, −7.74114271277496879280170273779, −6.98954755075212374936151715629, −4.36740576079605754911539238250, −3.56450683729495510949285530676, −2.30714960413351034597310421570, −1.23548968953120441800403208342, 1.06278345544151081211379090368, 2.56764840532515714929686137937, 3.81772987291375554962085126437, 5.66657848889535128818380000120, 6.62043604608160208352930742141, 8.225749393206241914144000861472, 8.505734170732484514363380865924, 9.077800331427824563861996548547, 10.39296963616183586975303034477, 11.36348172837691930440623195170

Graph of the ZZ-function along the critical line