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Properties

Label 2-147-147.101-c1-0-8
Degree 22
Conductor 147147
Sign 0.514−0.857i0.514 - 0.857i
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  + (1.58 + 0.620i)2-s + (0.433 + 1.67i)3-s + (0.647 + 0.601i)4-s + (0.761 + 0.519i)5-s + (−0.354 + 2.91i)6-s + (−0.654 − 2.56i)7-s + (−0.822 − 1.70i)8-s + (−2.62 + 1.45i)9-s + (0.881 + 1.29i)10-s + (−0.0278 − 0.184i)11-s + (−0.727 + 1.34i)12-s + (−2.33 + 1.86i)13-s + (0.556 − 4.45i)14-s + (−0.540 + 1.50i)15-s + (−0.372 − 4.97i)16-s + (3.71 + 1.14i)17-s + ⋯
L(s)  = 1  + (1.11 + 0.438i)2-s + (0.250 + 0.968i)3-s + (0.323 + 0.300i)4-s + (0.340 + 0.232i)5-s + (−0.144 + 1.19i)6-s + (−0.247 − 0.968i)7-s + (−0.290 − 0.603i)8-s + (−0.874 + 0.484i)9-s + (0.278 + 0.408i)10-s + (−0.00838 − 0.0556i)11-s + (−0.209 + 0.388i)12-s + (−0.648 + 0.517i)13-s + (0.148 − 1.19i)14-s + (−0.139 + 0.387i)15-s + (−0.0931 − 1.24i)16-s + (0.901 + 0.278i)17-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.514−0.857i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+1/2) L(s)=((0.514−0.857i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 0.514−0.857i0.514 - 0.857i
Analytic conductor: 1.173801.17380
Root analytic conductor: 1.083421.08342
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ147(101,⋅)\chi_{147} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :1/2), 0.514−0.857i)(2,\ 147,\ (\ :1/2),\ 0.514 - 0.857i)

Particular Values

L(1)L(1) ≈\approx 1.63459+0.925765i1.63459 + 0.925765i
L(12)L(\frac12) ≈\approx 1.63459+0.925765i1.63459 + 0.925765i
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+(−0.433−1.67i)T 1 + (-0.433 - 1.67i)T
7 1+(0.654+2.56i)T 1 + (0.654 + 2.56i)T
good2 1+(−1.58−0.620i)T+(1.46+1.36i)T2 1 + (-1.58 - 0.620i)T + (1.46 + 1.36i)T^{2}
5 1+(−0.761−0.519i)T+(1.82+4.65i)T2 1 + (-0.761 - 0.519i)T + (1.82 + 4.65i)T^{2}
11 1+(0.0278+0.184i)T+(−10.5+3.24i)T2 1 + (0.0278 + 0.184i)T + (-10.5 + 3.24i)T^{2}
13 1+(2.33−1.86i)T+(2.89−12.6i)T2 1 + (2.33 - 1.86i)T + (2.89 - 12.6i)T^{2}
17 1+(−3.71−1.14i)T+(14.0+9.57i)T2 1 + (-3.71 - 1.14i)T + (14.0 + 9.57i)T^{2}
19 1+(−2.14+1.24i)T+(9.5−16.4i)T2 1 + (-2.14 + 1.24i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.363−1.17i)T+(−19.0+12.9i)T2 1 + (-0.363 - 1.17i)T + (-19.0 + 12.9i)T^{2}
29 1+(9.18−2.09i)T+(26.1−12.5i)T2 1 + (9.18 - 2.09i)T + (26.1 - 12.5i)T^{2}
31 1+(−6.62−3.82i)T+(15.5+26.8i)T2 1 + (-6.62 - 3.82i)T + (15.5 + 26.8i)T^{2}
37 1+(−2.37+2.20i)T+(2.76−36.8i)T2 1 + (-2.37 + 2.20i)T + (2.76 - 36.8i)T^{2}
41 1+(5.40−2.60i)T+(25.5−32.0i)T2 1 + (5.40 - 2.60i)T + (25.5 - 32.0i)T^{2}
43 1+(−4.29−2.06i)T+(26.8+33.6i)T2 1 + (-4.29 - 2.06i)T + (26.8 + 33.6i)T^{2}
47 1+(3.79−9.66i)T+(−34.4−31.9i)T2 1 + (3.79 - 9.66i)T + (-34.4 - 31.9i)T^{2}
53 1+(1.45−1.56i)T+(−3.96−52.8i)T2 1 + (1.45 - 1.56i)T + (-3.96 - 52.8i)T^{2}
59 1+(−8.18+5.58i)T+(21.5−54.9i)T2 1 + (-8.18 + 5.58i)T + (21.5 - 54.9i)T^{2}
61 1+(3.96+4.27i)T+(−4.55+60.8i)T2 1 + (3.96 + 4.27i)T + (-4.55 + 60.8i)T^{2}
67 1+(−5.08+8.81i)T+(−33.5−58.0i)T2 1 + (-5.08 + 8.81i)T + (-33.5 - 58.0i)T^{2}
71 1+(0.367+0.0838i)T+(63.9+30.8i)T2 1 + (0.367 + 0.0838i)T + (63.9 + 30.8i)T^{2}
73 1+(7.84−3.08i)T+(53.5−49.6i)T2 1 + (7.84 - 3.08i)T + (53.5 - 49.6i)T^{2}
79 1+(−7.67−13.2i)T+(−39.5+68.4i)T2 1 + (-7.67 - 13.2i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.837−1.04i)T+(−18.4−80.9i)T2 1 + (0.837 - 1.04i)T + (-18.4 - 80.9i)T^{2}
89 1+(−5.96−0.899i)T+(85.0+26.2i)T2 1 + (-5.96 - 0.899i)T + (85.0 + 26.2i)T^{2}
97 1+5.60iT−97T2 1 + 5.60iT - 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

−13.64907019204422355414132255758, −12.50593905961764996709913624314, −11.20870690103644283038350475624, −10.00755739341221699766675500155, −9.474023773137531805851001397563, −7.72818248218421579730700029017, −6.47419162265056592883897678416, −5.27693865190300198193174730479, −4.25412162216836010876040076763, −3.19253298710051704439491001567, 2.20764313582445215297493365859, 3.34790411110413128544472933226, 5.28340042034895223060041009052, 5.93704481712643768470779217352, 7.54200176723334283656913741606, 8.664568245988073281066159616862, 9.788936122620001252935470007915, 11.62039339060623483297274133215, 12.05950033500824562908945551550, 13.00310413419353883274116252637

Graph of the ZZ-function along the critical line