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Properties

Label 2-147-21.20-c5-0-21
Degree 22
Conductor 147147
Sign −0.997−0.0755i-0.997 - 0.0755i
Analytic cond. 23.576423.5764
Root an. cond. 4.855554.85555
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  + 10.3i·2-s + (12.5 − 9.28i)3-s − 74.8·4-s + 31.7·5-s + (95.9 + 129. i)6-s − 442. i·8-s + (70.5 − 232. i)9-s + 328. i·10-s + 453. i·11-s + (−936. + 694. i)12-s + 551. i·13-s + (397. − 294. i)15-s + 2.17e3·16-s + 538.·17-s + (2.40e3 + 729. i)18-s + 1.36e3i·19-s + ⋯
L(s)  = 1  + 1.82i·2-s + (0.803 − 0.595i)3-s − 2.33·4-s + 0.567·5-s + (1.08 + 1.46i)6-s − 2.44i·8-s + (0.290 − 0.956i)9-s + 1.03i·10-s + 1.13i·11-s + (−1.87 + 1.39i)12-s + 0.905i·13-s + (0.456 − 0.338i)15-s + 2.12·16-s + 0.451·17-s + (1.74 + 0.530i)18-s + 0.868i·19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((−0.997−0.0755i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0755i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+5/2) L(s)=((−0.997−0.0755i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: −0.997−0.0755i-0.997 - 0.0755i
Analytic conductor: 23.576423.5764
Root analytic conductor: 4.855554.85555
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ147(146,⋅)\chi_{147} (146, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :5/2), −0.997−0.0755i)(2,\ 147,\ (\ :5/2),\ -0.997 - 0.0755i)

Particular Values

L(3)L(3) ≈\approx 2.0652561942.065256194
L(12)L(\frac12) ≈\approx 2.0652561942.065256194
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)
bad3 1+(−12.5+9.28i)T 1 + (-12.5 + 9.28i)T
7 1 1
good2 1−10.3iT−32T2 1 - 10.3iT - 32T^{2}
5 1−31.7T+3.12e3T2 1 - 31.7T + 3.12e3T^{2}
11 1−453.iT−1.61e5T2 1 - 453. iT - 1.61e5T^{2}
13 1−551.iT−3.71e5T2 1 - 551. iT - 3.71e5T^{2}
17 1−538.T+1.41e6T2 1 - 538.T + 1.41e6T^{2}
19 1−1.36e3iT−2.47e6T2 1 - 1.36e3iT - 2.47e6T^{2}
23 1−3.23e3iT−6.43e6T2 1 - 3.23e3iT - 6.43e6T^{2}
29 1−1.60e3iT−2.05e7T2 1 - 1.60e3iT - 2.05e7T^{2}
31 1−7.06e3iT−2.86e7T2 1 - 7.06e3iT - 2.86e7T^{2}
37 1+1.95e3T+6.93e7T2 1 + 1.95e3T + 6.93e7T^{2}
41 1−1.80e3T+1.15e8T2 1 - 1.80e3T + 1.15e8T^{2}
43 1−7.88e3T+1.47e8T2 1 - 7.88e3T + 1.47e8T^{2}
47 1+6.09e3T+2.29e8T2 1 + 6.09e3T + 2.29e8T^{2}
53 1+1.41e4iT−4.18e8T2 1 + 1.41e4iT - 4.18e8T^{2}
59 1−1.69e4T+7.14e8T2 1 - 1.69e4T + 7.14e8T^{2}
61 1+2.97e4iT−8.44e8T2 1 + 2.97e4iT - 8.44e8T^{2}
67 1−1.36e4T+1.35e9T2 1 - 1.36e4T + 1.35e9T^{2}
71 1−3.13e4iT−1.80e9T2 1 - 3.13e4iT - 1.80e9T^{2}
73 1+9.82e3iT−2.07e9T2 1 + 9.82e3iT - 2.07e9T^{2}
79 1−9.97e4T+3.07e9T2 1 - 9.97e4T + 3.07e9T^{2}
83 1+7.95e4T+3.93e9T2 1 + 7.95e4T + 3.93e9T^{2}
89 1−953.T+5.58e9T2 1 - 953.T + 5.58e9T^{2}
97 1−1.15e5iT−8.58e9T2 1 - 1.15e5iT - 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

−13.11242847235928724735000457446, −12.14422388142696222982511136741, −9.840315142216608631546996718557, −9.261714412127142254704397946972, −8.130111738427793388703526316662, −7.27067183446692306916048016564, −6.49348034959658641077974256776, −5.29563066432323496730273193861, −3.86560714547947192078378353799, −1.67259314751575459525657570244, 0.65387480604710698956255772656, 2.29843476779164296233528912695, 3.14265616060194153386868106795, 4.30258598551283273855965262265, 5.61818714256207048403805769753, 8.066264798800074743763607551732, 8.962405921018631973178574005881, 9.837205170684750236540164313221, 10.58734574080317143693081177937, 11.38169691268023193830332433379

Graph of the ZZ-function along the critical line