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Properties

Label 2-147-147.104-c1-0-1
Degree 22
Conductor 147147
Sign 0.180+0.983i0.180 + 0.983i
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  + (−1.77 − 1.41i)2-s + (−1.67 − 0.435i)3-s + (0.698 + 3.05i)4-s + (3.64 + 1.75i)5-s + (2.35 + 3.14i)6-s + (0.439 − 2.60i)7-s + (1.11 − 2.32i)8-s + (2.62 + 1.45i)9-s + (−3.98 − 8.27i)10-s + (−0.206 − 0.164i)11-s + (0.160 − 5.43i)12-s + (−0.541 − 0.431i)13-s + (−4.46 + 4.00i)14-s + (−5.35 − 4.53i)15-s + (0.390 − 0.188i)16-s + (0.756 − 3.31i)17-s + ⋯
L(s)  = 1  + (−1.25 − 0.999i)2-s + (−0.967 − 0.251i)3-s + (0.349 + 1.52i)4-s + (1.63 + 0.785i)5-s + (0.961 + 1.28i)6-s + (0.166 − 0.986i)7-s + (0.395 − 0.820i)8-s + (0.873 + 0.486i)9-s + (−1.25 − 2.61i)10-s + (−0.0621 − 0.0495i)11-s + (0.0464 − 1.56i)12-s + (−0.150 − 0.119i)13-s + (−1.19 + 1.06i)14-s + (−1.38 − 1.17i)15-s + (0.0976 − 0.0470i)16-s + (0.183 − 0.803i)17-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.180+0.983i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+1/2) L(s)=((0.180+0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 0.180+0.983i0.180 + 0.983i
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), 0.180+0.983i)(2,\ 147,\ (\ :1/2),\ 0.180 + 0.983i)

Particular Values

L(1)L(1) ≈\approx 0.459970−0.383193i0.459970 - 0.383193i
L(12)L(\frac12) ≈\approx 0.459970−0.383193i0.459970 - 0.383193i
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+(1.67+0.435i)T 1 + (1.67 + 0.435i)T
7 1+(−0.439+2.60i)T 1 + (-0.439 + 2.60i)T
good2 1+(1.77+1.41i)T+(0.445+1.94i)T2 1 + (1.77 + 1.41i)T + (0.445 + 1.94i)T^{2}
5 1+(−3.64−1.75i)T+(3.11+3.90i)T2 1 + (-3.64 - 1.75i)T + (3.11 + 3.90i)T^{2}
11 1+(0.206+0.164i)T+(2.44+10.7i)T2 1 + (0.206 + 0.164i)T + (2.44 + 10.7i)T^{2}
13 1+(0.541+0.431i)T+(2.89+12.6i)T2 1 + (0.541 + 0.431i)T + (2.89 + 12.6i)T^{2}
17 1+(−0.756+3.31i)T+(−15.3−7.37i)T2 1 + (-0.756 + 3.31i)T + (-15.3 - 7.37i)T^{2}
19 1+2.65iT−19T2 1 + 2.65iT - 19T^{2}
23 1+(−7.19+1.64i)T+(20.7−9.97i)T2 1 + (-7.19 + 1.64i)T + (20.7 - 9.97i)T^{2}
29 1+(0.664+0.151i)T+(26.1+12.5i)T2 1 + (0.664 + 0.151i)T + (26.1 + 12.5i)T^{2}
31 1−4.03iT−31T2 1 - 4.03iT - 31T^{2}
37 1+(−1.08+4.76i)T+(−33.3−16.0i)T2 1 + (-1.08 + 4.76i)T + (-33.3 - 16.0i)T^{2}
41 1+(−4.22−2.03i)T+(25.5+32.0i)T2 1 + (-4.22 - 2.03i)T + (25.5 + 32.0i)T^{2}
43 1+(1.86−0.899i)T+(26.8−33.6i)T2 1 + (1.86 - 0.899i)T + (26.8 - 33.6i)T^{2}
47 1+(4.97−6.24i)T+(−10.4−45.8i)T2 1 + (4.97 - 6.24i)T + (-10.4 - 45.8i)T^{2}
53 1+(1.53−0.349i)T+(47.7−22.9i)T2 1 + (1.53 - 0.349i)T + (47.7 - 22.9i)T^{2}
59 1+(10.5−5.09i)T+(36.7−46.1i)T2 1 + (10.5 - 5.09i)T + (36.7 - 46.1i)T^{2}
61 1+(−2.91−0.664i)T+(54.9+26.4i)T2 1 + (-2.91 - 0.664i)T + (54.9 + 26.4i)T^{2}
67 1−5.56T+67T2 1 - 5.56T + 67T^{2}
71 1+(−10.8+2.47i)T+(63.9−30.8i)T2 1 + (-10.8 + 2.47i)T + (63.9 - 30.8i)T^{2}
73 1+(9.38−7.48i)T+(16.2−71.1i)T2 1 + (9.38 - 7.48i)T + (16.2 - 71.1i)T^{2}
79 1−4.72T+79T2 1 - 4.72T + 79T^{2}
83 1+(2.28+2.86i)T+(−18.4+80.9i)T2 1 + (2.28 + 2.86i)T + (-18.4 + 80.9i)T^{2}
89 1+(2.41+3.03i)T+(−19.8+86.7i)T2 1 + (2.41 + 3.03i)T + (-19.8 + 86.7i)T^{2}
97 1+0.586iT−97T2 1 + 0.586iT - 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

−12.65829523542882954618302160171, −11.17974722852040292050041181746, −10.82450800690396756071891595672, −9.988893568504623149229820652616, −9.256684217456741127396984790945, −7.47850111933171681273120138408, −6.61330400117859448953026358395, −5.15756709802532620682542161999, −2.74061020818602123020089892172, −1.22841610135225942240560001971, 1.54032320403391969970080419936, 5.14463394292722040477355056454, 5.82543422305809617057045249429, 6.64020097691445456587901214307, 8.265566045447944367037765153018, 9.316211859008255204849971158933, 9.760984499765504508079816506876, 10.83518480767434535866235723260, 12.32172422469965941865771893695, 13.14496132735445674315206272658

Graph of the ZZ-function along the critical line