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Properties

Label 2-147-147.104-c1-0-3
Degree 22
Conductor 147147
Sign 0.433+0.901i0.433 + 0.901i
Analytic cond. 1.173801.17380
Root an. cond. 1.083421.08342
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.95 − 1.56i)2-s + (1.73 − 0.0781i)3-s + (0.951 + 4.16i)4-s + (−0.242 − 0.116i)5-s + (−3.51 − 2.54i)6-s + (2.60 + 0.481i)7-s + (2.47 − 5.13i)8-s + (2.98 − 0.270i)9-s + (0.292 + 0.607i)10-s + (2.92 + 2.33i)11-s + (1.97 + 7.13i)12-s + (−4.15 − 3.31i)13-s + (−4.34 − 5.00i)14-s + (−0.428 − 0.183i)15-s + (−5.16 + 2.48i)16-s + (0.256 − 1.12i)17-s + ⋯
L(s)  = 1  + (−1.38 − 1.10i)2-s + (0.998 − 0.0451i)3-s + (0.475 + 2.08i)4-s + (−0.108 − 0.0522i)5-s + (−1.43 − 1.04i)6-s + (0.983 + 0.181i)7-s + (0.874 − 1.81i)8-s + (0.995 − 0.0901i)9-s + (0.0925 + 0.192i)10-s + (0.881 + 0.703i)11-s + (0.569 + 2.06i)12-s + (−1.15 − 0.919i)13-s + (−1.16 − 1.33i)14-s + (−0.110 − 0.0472i)15-s + (−1.29 + 0.621i)16-s + (0.0621 − 0.272i)17-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.433+0.901i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+1/2) L(s)=((0.433+0.901i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 0.433+0.901i0.433 + 0.901i
Analytic conductor: 1.173801.17380
Root analytic conductor: 1.083421.08342
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ147(104,⋅)\chi_{147} (104, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :1/2), 0.433+0.901i)(2,\ 147,\ (\ :1/2),\ 0.433 + 0.901i)

Particular Values

L(1)L(1) ≈\approx 0.718747−0.451743i0.718747 - 0.451743i
L(12)L(\frac12) ≈\approx 0.718747−0.451743i0.718747 - 0.451743i
L(32)L(\frac{3}{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+(−1.73+0.0781i)T 1 + (-1.73 + 0.0781i)T
7 1+(−2.60−0.481i)T 1 + (-2.60 - 0.481i)T
good2 1+(1.95+1.56i)T+(0.445+1.94i)T2 1 + (1.95 + 1.56i)T + (0.445 + 1.94i)T^{2}
5 1+(0.242+0.116i)T+(3.11+3.90i)T2 1 + (0.242 + 0.116i)T + (3.11 + 3.90i)T^{2}
11 1+(−2.92−2.33i)T+(2.44+10.7i)T2 1 + (-2.92 - 2.33i)T + (2.44 + 10.7i)T^{2}
13 1+(4.15+3.31i)T+(2.89+12.6i)T2 1 + (4.15 + 3.31i)T + (2.89 + 12.6i)T^{2}
17 1+(−0.256+1.12i)T+(−15.3−7.37i)T2 1 + (-0.256 + 1.12i)T + (-15.3 - 7.37i)T^{2}
19 1+2.04iT−19T2 1 + 2.04iT - 19T^{2}
23 1+(5.12−1.16i)T+(20.7−9.97i)T2 1 + (5.12 - 1.16i)T + (20.7 - 9.97i)T^{2}
29 1+(−4.08−0.932i)T+(26.1+12.5i)T2 1 + (-4.08 - 0.932i)T + (26.1 + 12.5i)T^{2}
31 1−6.91iT−31T2 1 - 6.91iT - 31T^{2}
37 1+(1.90−8.32i)T+(−33.3−16.0i)T2 1 + (1.90 - 8.32i)T + (-33.3 - 16.0i)T^{2}
41 1+(−0.0133−0.00643i)T+(25.5+32.0i)T2 1 + (-0.0133 - 0.00643i)T + (25.5 + 32.0i)T^{2}
43 1+(−0.419+0.202i)T+(26.8−33.6i)T2 1 + (-0.419 + 0.202i)T + (26.8 - 33.6i)T^{2}
47 1+(−6.73+8.44i)T+(−10.4−45.8i)T2 1 + (-6.73 + 8.44i)T + (-10.4 - 45.8i)T^{2}
53 1+(6.59−1.50i)T+(47.7−22.9i)T2 1 + (6.59 - 1.50i)T + (47.7 - 22.9i)T^{2}
59 1+(9.03−4.35i)T+(36.7−46.1i)T2 1 + (9.03 - 4.35i)T + (36.7 - 46.1i)T^{2}
61 1+(9.35+2.13i)T+(54.9+26.4i)T2 1 + (9.35 + 2.13i)T + (54.9 + 26.4i)T^{2}
67 1+1.25T+67T2 1 + 1.25T + 67T^{2}
71 1+(13.9−3.18i)T+(63.9−30.8i)T2 1 + (13.9 - 3.18i)T + (63.9 - 30.8i)T^{2}
73 1+(3.13−2.49i)T+(16.2−71.1i)T2 1 + (3.13 - 2.49i)T + (16.2 - 71.1i)T^{2}
79 1−2.82T+79T2 1 - 2.82T + 79T^{2}
83 1+(2.80+3.51i)T+(−18.4+80.9i)T2 1 + (2.80 + 3.51i)T + (-18.4 + 80.9i)T^{2}
89 1+(−1.44−1.80i)T+(−19.8+86.7i)T2 1 + (-1.44 - 1.80i)T + (-19.8 + 86.7i)T^{2}
97 1+2.81iT−97T2 1 + 2.81iT - 97T^{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

−12.21638007456718437925595605872, −12.03244091956498871376672478135, −10.48468932135696850146315014485, −9.817917899151271075409819640672, −8.828036749720095662883715691788, −8.025579996379112688272862090680, −7.21436638459130427509131223161, −4.50760352388890182920776962522, −2.88802731217797671317387012021, −1.66060655812278351465315979367, 1.77264683538807771712933060735, 4.28084598329985614258793978224, 6.07932932301345865215287893608, 7.39480444817131043057556824174, 7.945479207033334408304553725337, 8.980988457127131446529045697641, 9.639262954668740106379529385521, 10.78064240468147479329387255713, 12.04698385237093749246608557574, 13.98270889120030855595442871838

Graph of the ZZ-function along the critical line