↑

Properties

Label 2-147-7.2-c5-0-30
Degree 22
Conductor 147147
Sign −0.991−0.126i-0.991 - 0.126i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.59 − 7.95i)2-s + (4.5 + 7.79i)3-s + (−26.1 − 45.2i)4-s + (11.0 − 19.1i)5-s + 82.6·6-s − 186.·8-s + (−40.5 + 70.1i)9-s + (−101. − 175. i)10-s + (−208. − 360. i)11-s + (235. − 407. i)12-s − 797.·13-s + 198.·15-s + (−18.6 + 32.3i)16-s + (−687. − 1.19e3i)17-s + (371. + 644. i)18-s + (1.15e3 − 2.00e3i)19-s + ⋯
L(s)  = 1  + (0.811 − 1.40i)2-s + (0.288 + 0.499i)3-s + (−0.817 − 1.41i)4-s + (0.197 − 0.341i)5-s + 0.937·6-s − 1.02·8-s + (−0.166 + 0.288i)9-s + (−0.320 − 0.554i)10-s + (−0.519 − 0.899i)11-s + (0.471 − 0.817i)12-s − 1.30·13-s + 0.227·15-s + (−0.0182 + 0.0315i)16-s + (−0.577 − 0.999i)17-s + (0.270 + 0.468i)18-s + (0.734 − 1.27i)19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((−0.991−0.126i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+5/2) L(s)=((−0.991−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: −0.991−0.126i-0.991 - 0.126i
Analytic conductor: 23.576423.5764
Root analytic conductor: 4.855554.85555
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ147(79,⋅)\chi_{147} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :5/2), −0.991−0.126i)(2,\ 147,\ (\ :5/2),\ -0.991 - 0.126i)

Particular Values

L(3)L(3) ≈\approx 2.3346813742.334681374
L(12)L(\frac12) ≈\approx 2.3346813742.334681374
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+(−4.5−7.79i)T 1 + (-4.5 - 7.79i)T
7 1 1
good2 1+(−4.59+7.95i)T+(−16−27.7i)T2 1 + (-4.59 + 7.95i)T + (-16 - 27.7i)T^{2}
5 1+(−11.0+19.1i)T+(−1.56e3−2.70e3i)T2 1 + (-11.0 + 19.1i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(208.+360.i)T+(−8.05e4+1.39e5i)T2 1 + (208. + 360. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+797.T+3.71e5T2 1 + 797.T + 3.71e5T^{2}
17 1+(687.+1.19e3i)T+(−7.09e5+1.22e6i)T2 1 + (687. + 1.19e3i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−1.15e3+2.00e3i)T+(−1.23e6−2.14e6i)T2 1 + (-1.15e3 + 2.00e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−477.+827.i)T+(−3.21e6−5.57e6i)T2 1 + (-477. + 827. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+7.03e3T+2.05e7T2 1 + 7.03e3T + 2.05e7T^{2}
31 1+(−630.−1.09e3i)T+(−1.43e7+2.47e7i)T2 1 + (-630. - 1.09e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(4.88e3−8.46e3i)T+(−3.46e7−6.00e7i)T2 1 + (4.88e3 - 8.46e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−5.40e3T+1.15e8T2 1 - 5.40e3T + 1.15e8T^{2}
43 1−1.96e4T+1.47e8T2 1 - 1.96e4T + 1.47e8T^{2}
47 1+(−1.02e3+1.78e3i)T+(−1.14e8−1.98e8i)T2 1 + (-1.02e3 + 1.78e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(9.01e3+1.56e4i)T+(−2.09e8+3.62e8i)T2 1 + (9.01e3 + 1.56e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−3.71e3−6.43e3i)T+(−3.57e8+6.19e8i)T2 1 + (-3.71e3 - 6.43e3i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−1.74e3+3.02e3i)T+(−4.22e8−7.31e8i)T2 1 + (-1.74e3 + 3.02e3i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(7.92e3+1.37e4i)T+(−6.75e8+1.16e9i)T2 1 + (7.92e3 + 1.37e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−5.81e4T+1.80e9T2 1 - 5.81e4T + 1.80e9T^{2}
73 1+(−1.95e4−3.38e4i)T+(−1.03e9+1.79e9i)T2 1 + (-1.95e4 - 3.38e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(4.88e3−8.45e3i)T+(−1.53e9−2.66e9i)T2 1 + (4.88e3 - 8.45e3i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−7.03e4T+3.93e9T2 1 - 7.03e4T + 3.93e9T^{2}
89 1+(−7.21e4+1.24e5i)T+(−2.79e9−4.83e9i)T2 1 + (-7.21e4 + 1.24e5i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−7.93e4T+8.58e9T2 1 - 7.93e4T + 8.58e9T^{2}
show more
show less
   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

−11.49438990463545628546477636352, −10.91281227628661464338405879748, −9.720271996588236137951028684577, −9.050706312306071607252860432709, −7.36801561970985846652697332323, −5.32577318534173093049250012105, −4.72957975982543698217639054564, −3.23844859942235457790333170546, −2.36397423999022205422348389040, −0.55243391021753936573227649399, 2.17325847057059899031331605466, 3.91480400933301333557298442757, 5.21881478637155298753839855213, 6.24064957355662710594290625112, 7.38598943051586544447580285725, 7.80361885954052505471196119962, 9.314294834162840189750595297530, 10.57305772372926912738942836236, 12.34487474720154039834614486751, 12.79816465498666765827223931725

Graph of the ZZ-function along the critical line