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Properties

Label 2-147-7.2-c1-0-5
Degree 22
Conductor 147147
Sign −0.0725+0.997i-0.0725 + 0.997i
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  + (1.20 − 2.09i)2-s + (0.5 + 0.866i)3-s + (−1.91 − 3.31i)4-s + (0.292 − 0.507i)5-s + 2.41·6-s − 4.41·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (1 + 1.73i)11-s + (1.91 − 3.31i)12-s − 5.41·13-s + 0.585·15-s + (−1.49 + 2.59i)16-s + (3.12 + 5.40i)17-s + (1.20 + 2.09i)18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (0.853 − 1.47i)2-s + (0.288 + 0.499i)3-s + (−0.957 − 1.65i)4-s + (0.130 − 0.226i)5-s + 0.985·6-s − 1.56·8-s + (−0.166 + 0.288i)9-s + (−0.223 − 0.387i)10-s + (0.301 + 0.522i)11-s + (0.552 − 0.957i)12-s − 1.50·13-s + 0.151·15-s + (−0.374 + 0.649i)16-s + (0.757 + 1.31i)17-s + (0.284 + 0.492i)18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((−0.0725+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+1/2) L(s)=((−0.0725+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: −0.0725+0.997i-0.0725 + 0.997i
Analytic conductor: 1.173801.17380
Root analytic conductor: 1.083421.08342
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ147(79,⋅)\chi_{147} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :1/2), −0.0725+0.997i)(2,\ 147,\ (\ :1/2),\ -0.0725 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 1.15995−1.24740i1.15995 - 1.24740i
L(12)L(\frac12) ≈\approx 1.15995−1.24740i1.15995 - 1.24740i
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1 1
good2 1+(−1.20+2.09i)T+(−1−1.73i)T2 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2}
5 1+(−0.292+0.507i)T+(−2.5−4.33i)T2 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+5.41T+13T2 1 + 5.41T + 13T^{2}
17 1+(−3.12−5.40i)T+(−8.5+14.7i)T2 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.41+2.44i)T+(−9.5−16.4i)T2 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.82−3.16i)T+(−11.5−19.9i)T2 1 + (1.82 - 3.16i)T + (-11.5 - 19.9i)T^{2}
29 1+1.17T+29T2 1 + 1.17T + 29T^{2}
31 1+(3.41+5.91i)T+(−15.5+26.8i)T2 1 + (3.41 + 5.91i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2+3.46i)T+(−18.5−32.0i)T2 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}
41 1−2.24T+41T2 1 - 2.24T + 41T^{2}
43 1+5.65T+43T2 1 + 5.65T + 43T^{2}
47 1+(−1.41+2.44i)T+(−23.5−40.7i)T2 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1−1.73i)T+(−26.5+45.8i)T2 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2}
59 1+(3.41+5.91i)T+(−29.5+51.0i)T2 1 + (3.41 + 5.91i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.87+3.25i)T+(−30.5−52.8i)T2 1 + (-1.87 + 3.25i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.82+4.89i)T+(−33.5+58.0i)T2 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2}
71 1+13.3T+71T2 1 + 13.3T + 71T^{2}
73 1+(2.94+5.10i)T+(−36.5+63.2i)T2 1 + (2.94 + 5.10i)T + (-36.5 + 63.2i)T^{2}
79 1+(1.17−2.02i)T+(−39.5−68.4i)T2 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2}
83 1−15.3T+83T2 1 - 15.3T + 83T^{2}
89 1+(2.87−4.98i)T+(−44.5−77.0i)T2 1 + (2.87 - 4.98i)T + (-44.5 - 77.0i)T^{2}
97 1+5.41T+97T2 1 + 5.41T + 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

−12.66702220215125935320562188260, −11.92217519573507199466513811231, −10.87811590835846789043656763950, −9.889510928048099504963370216341, −9.311656244740961829791763617556, −7.57433717323487988304552963124, −5.57176317418056938817905304053, −4.57043138743411482840729084786, −3.43581297509463027675472343068, −1.98677109128099668675269595975, 3.03684330363718662259261721206, 4.69307314737029500098423310459, 5.80647170531078770796941100252, 6.94583933269027828633817373627, 7.61294315908616873244959779219, 8.727657081669744181563898226769, 10.04692161610703631011305794187, 11.87814090679738796004096596894, 12.58572258318403150468513277789, 13.75230583028398820210289807479

Graph of the ZZ-function along the critical line