Properties

Label 2-200-40.29-c5-0-38
Degree 22
Conductor 200200
Sign 0.9650.261i0.965 - 0.261i
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  + (−0.375 + 5.64i)2-s − 6.67·3-s + (−31.7 − 4.24i)4-s + (2.50 − 37.6i)6-s + 38.2i·7-s + (35.8 − 177. i)8-s − 198.·9-s + 491. i·11-s + (211. + 28.3i)12-s − 956.·13-s + (−216. − 14.3i)14-s + (988. + 269. i)16-s − 339. i·17-s + (74.5 − 1.12e3i)18-s − 1.86e3i·19-s + ⋯
L(s)  = 1  + (−0.0664 + 0.997i)2-s − 0.428·3-s + (−0.991 − 0.132i)4-s + (0.0284 − 0.427i)6-s + 0.295i·7-s + (0.198 − 0.980i)8-s − 0.816·9-s + 1.22i·11-s + (0.424 + 0.0567i)12-s − 1.56·13-s + (−0.294 − 0.0196i)14-s + (0.964 + 0.262i)16-s − 0.285i·17-s + (0.0542 − 0.814i)18-s − 1.18i·19-s + ⋯

Functional equation

Λ(s)=(200s/2ΓC(s)L(s)=((0.9650.261i)Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2ΓC(s+5/2)L(s)=((0.9650.261i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23522^{3} \cdot 5^{2}
Sign: 0.9650.261i0.965 - 0.261i
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), 0.9650.261i)(2,\ 200,\ (\ :5/2),\ 0.965 - 0.261i)

Particular Values

L(3)L(3) \approx 0.75913837300.7591383730
L(12)L(\frac12) \approx 0.75913837300.7591383730
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+(0.3755.64i)T 1 + (0.375 - 5.64i)T
5 1 1
good3 1+6.67T+243T2 1 + 6.67T + 243T^{2}
7 138.2iT1.68e4T2 1 - 38.2iT - 1.68e4T^{2}
11 1491.iT1.61e5T2 1 - 491. iT - 1.61e5T^{2}
13 1+956.T+3.71e5T2 1 + 956.T + 3.71e5T^{2}
17 1+339.iT1.41e6T2 1 + 339. iT - 1.41e6T^{2}
19 1+1.86e3iT2.47e6T2 1 + 1.86e3iT - 2.47e6T^{2}
23 1+1.99e3iT6.43e6T2 1 + 1.99e3iT - 6.43e6T^{2}
29 13.57e3iT2.05e7T2 1 - 3.57e3iT - 2.05e7T^{2}
31 17.71e3T+2.86e7T2 1 - 7.71e3T + 2.86e7T^{2}
37 1+2.83e3T+6.93e7T2 1 + 2.83e3T + 6.93e7T^{2}
41 11.06e4T+1.15e8T2 1 - 1.06e4T + 1.15e8T^{2}
43 1+2.05e4T+1.47e8T2 1 + 2.05e4T + 1.47e8T^{2}
47 1756.iT2.29e8T2 1 - 756. iT - 2.29e8T^{2}
53 13.16e4T+4.18e8T2 1 - 3.16e4T + 4.18e8T^{2}
59 14.91e3iT7.14e8T2 1 - 4.91e3iT - 7.14e8T^{2}
61 1+2.14e4iT8.44e8T2 1 + 2.14e4iT - 8.44e8T^{2}
67 16.81e3T+1.35e9T2 1 - 6.81e3T + 1.35e9T^{2}
71 1+1.11e4T+1.80e9T2 1 + 1.11e4T + 1.80e9T^{2}
73 17.30e3iT2.07e9T2 1 - 7.30e3iT - 2.07e9T^{2}
79 12.35e4T+3.07e9T2 1 - 2.35e4T + 3.07e9T^{2}
83 12.37e4T+3.93e9T2 1 - 2.37e4T + 3.93e9T^{2}
89 11.25e5T+5.58e9T2 1 - 1.25e5T + 5.58e9T^{2}
97 1+1.26e5iT8.58e9T2 1 + 1.26e5iT - 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

−11.87415535647846747253727649367, −10.39865433192023626712907947567, −9.496023013761084664166526700972, −8.544193261175441980609889723112, −7.32156328774341323869785147008, −6.58429690226209908800585505431, −5.21650345702515544607824763872, −4.63442241584018000054133859795, −2.59779569765968550260995714071, −0.37183059294762901705913955112, 0.78445078425356259116948085501, 2.48224354329757778392844375436, 3.67012567046756779811012318896, 5.05992529777862055400929135655, 6.02766796261097661736438691968, 7.76408294899487113054827498475, 8.665127014216867801376370476799, 9.875601006756145272903865277943, 10.59580864176228426389987983193, 11.72241866606345609770693659914

Graph of the ZZ-function along the critical line