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Properties

Label 2-147-7.4-c5-0-25
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 − 2.37i)2-s + (4.5 − 7.79i)3-s + (12.2 − 21.1i)4-s + (29.1 + 50.5i)5-s − 24.7·6-s − 155.·8-s + (−40.5 − 70.1i)9-s + (80.2 − 138. i)10-s + (−8.71 + 15.0i)11-s + (−110. − 190. i)12-s − 889.·13-s + 525.·15-s + (−178. − 308. i)16-s + (513. − 889. i)17-s + (−111. + 192. i)18-s + (−869. − 1.50e3i)19-s + ⋯
L(s)  = 1  + (−0.242 − 0.420i)2-s + (0.288 − 0.499i)3-s + (0.381 − 0.661i)4-s + (0.522 + 0.904i)5-s − 0.280·6-s − 0.856·8-s + (−0.166 − 0.288i)9-s + (0.253 − 0.439i)10-s + (−0.0217 + 0.0376i)11-s + (−0.220 − 0.381i)12-s − 1.46·13-s + 0.602·15-s + (−0.173 − 0.301i)16-s + (0.430 − 0.746i)17-s + (−0.0809 + 0.140i)18-s + (−0.552 − 0.957i)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(67,⋅)\chi_{147} (67, \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 1.2263517681.226351768
L(12)L(\frac12) ≈\approx 1.2263517681.226351768
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+(1.37+2.37i)T+(−16+27.7i)T2 1 + (1.37 + 2.37i)T + (-16 + 27.7i)T^{2}
5 1+(−29.1−50.5i)T+(−1.56e3+2.70e3i)T2 1 + (-29.1 - 50.5i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(8.71−15.0i)T+(−8.05e4−1.39e5i)T2 1 + (8.71 - 15.0i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+889.T+3.71e5T2 1 + 889.T + 3.71e5T^{2}
17 1+(−513.+889.i)T+(−7.09e5−1.22e6i)T2 1 + (-513. + 889. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(869.+1.50e3i)T+(−1.23e6+2.14e6i)T2 1 + (869. + 1.50e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(1.96e3+3.40e3i)T+(−3.21e6+5.57e6i)T2 1 + (1.96e3 + 3.40e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1−5.63e3T+2.05e7T2 1 - 5.63e3T + 2.05e7T^{2}
31 1+(1.54e3−2.68e3i)T+(−1.43e7−2.47e7i)T2 1 + (1.54e3 - 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(2.51e3+4.35e3i)T+(−3.46e7+6.00e7i)T2 1 + (2.51e3 + 4.35e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+1.83e4T+1.15e8T2 1 + 1.83e4T + 1.15e8T^{2}
43 1+1.63e3T+1.47e8T2 1 + 1.63e3T + 1.47e8T^{2}
47 1+(4.80e3+8.31e3i)T+(−1.14e8+1.98e8i)T2 1 + (4.80e3 + 8.31e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−1.16e4+2.01e4i)T+(−2.09e8−3.62e8i)T2 1 + (-1.16e4 + 2.01e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(1.80e3−3.12e3i)T+(−3.57e8−6.19e8i)T2 1 + (1.80e3 - 3.12e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−1.14e4−1.98e4i)T+(−4.22e8+7.31e8i)T2 1 + (-1.14e4 - 1.98e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(2.35e4−4.07e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.35e4 - 4.07e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+1.59e3T+1.80e9T2 1 + 1.59e3T + 1.80e9T^{2}
73 1+(−2.96e3+5.13e3i)T+(−1.03e9−1.79e9i)T2 1 + (-2.96e3 + 5.13e3i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−4.42e4−7.66e4i)T+(−1.53e9+2.66e9i)T2 1 + (-4.42e4 - 7.66e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−9.58e4T+3.93e9T2 1 - 9.58e4T + 3.93e9T^{2}
89 1+(2.32e4+4.02e4i)T+(−2.79e9+4.83e9i)T2 1 + (2.32e4 + 4.02e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−7.59e4T+8.58e9T2 1 - 7.59e4T + 8.58e9T^{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

−11.65795208804143195692399580628, −10.39754567553760308304574113805, −9.948875825113031802585893931943, −8.669227509482487410975891443721, −7.07690440206091156886222192034, −6.51151774398990261191186259113, −5.06574526085432483642138370971, −2.81465718732409702506243497772, −2.15694558860676772556769789913, −0.38892228681373068113390755483, 1.93027652560405659351694265794, 3.48683772359282345589540075745, 4.92712954383861458374969040833, 6.13261458695421887491243551211, 7.61166578529036843484819667717, 8.403104705221070633884655646624, 9.439625051052719463334890828476, 10.28450796188846584782869603393, 11.89523144101434572779430449204, 12.49202621023790703216079888887

Graph of the ZZ-function along the critical line