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Properties

Label 2-147-7.4-c5-0-28
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  + (0.395 + 0.684i)2-s + (4.5 − 7.79i)3-s + (15.6 − 27.1i)4-s + (−52.0 − 90.2i)5-s + 7.11·6-s + 50.1·8-s + (−40.5 − 70.1i)9-s + (41.1 − 71.3i)10-s + (248. − 430. i)11-s + (−141. − 244. i)12-s + 206.·13-s − 937.·15-s + (−482. − 835. i)16-s + (31.5 − 54.6i)17-s + (32.0 − 55.4i)18-s + (661. + 1.14e3i)19-s + ⋯
L(s)  = 1  + (0.0698 + 0.121i)2-s + (0.288 − 0.499i)3-s + (0.490 − 0.849i)4-s + (−0.931 − 1.61i)5-s + 0.0807·6-s + 0.276·8-s + (−0.166 − 0.288i)9-s + (0.130 − 0.225i)10-s + (0.620 − 1.07i)11-s + (−0.283 − 0.490i)12-s + 0.338·13-s − 1.07·15-s + (−0.470 − 0.815i)16-s + (0.0265 − 0.0459i)17-s + (0.0232 − 0.0403i)18-s + (0.420 + 0.728i)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 1.8674068221.867406822
L(12)L(\frac12) ≈\approx 1.8674068221.867406822
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+(−0.395−0.684i)T+(−16+27.7i)T2 1 + (-0.395 - 0.684i)T + (-16 + 27.7i)T^{2}
5 1+(52.0+90.2i)T+(−1.56e3+2.70e3i)T2 1 + (52.0 + 90.2i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−248.+430.i)T+(−8.05e4−1.39e5i)T2 1 + (-248. + 430. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−206.T+3.71e5T2 1 - 206.T + 3.71e5T^{2}
17 1+(−31.5+54.6i)T+(−7.09e5−1.22e6i)T2 1 + (-31.5 + 54.6i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−661.−1.14e3i)T+(−1.23e6+2.14e6i)T2 1 + (-661. - 1.14e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−97.2−168.i)T+(−3.21e6+5.57e6i)T2 1 + (-97.2 - 168. i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1−4.32e3T+2.05e7T2 1 - 4.32e3T + 2.05e7T^{2}
31 1+(3.76e3−6.51e3i)T+(−1.43e7−2.47e7i)T2 1 + (3.76e3 - 6.51e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(5.17e3+8.96e3i)T+(−3.46e7+6.00e7i)T2 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−4.18e3T+1.15e8T2 1 - 4.18e3T + 1.15e8T^{2}
43 1−5.96e3T+1.47e8T2 1 - 5.96e3T + 1.47e8T^{2}
47 1+(2.19e3+3.80e3i)T+(−1.14e8+1.98e8i)T2 1 + (2.19e3 + 3.80e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(8.89e3−1.54e4i)T+(−2.09e8−3.62e8i)T2 1 + (8.89e3 - 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−1.75e3+3.03e3i)T+(−3.57e8−6.19e8i)T2 1 + (-1.75e3 + 3.03e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(5.31e3+9.20e3i)T+(−4.22e8+7.31e8i)T2 1 + (5.31e3 + 9.20e3i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(−6.63e3+1.14e4i)T+(−6.75e8−1.16e9i)T2 1 + (-6.63e3 + 1.14e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−3.88e4T+1.80e9T2 1 - 3.88e4T + 1.80e9T^{2}
73 1+(−1.56e4+2.71e4i)T+(−1.03e9−1.79e9i)T2 1 + (-1.56e4 + 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(1.97e4+3.42e4i)T+(−1.53e9+2.66e9i)T2 1 + (1.97e4 + 3.42e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−1.02e5T+3.93e9T2 1 - 1.02e5T + 3.93e9T^{2}
89 1+(5.64e4+9.77e4i)T+(−2.79e9+4.83e9i)T2 1 + (5.64e4 + 9.77e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+3.03e4T+8.58e9T2 1 + 3.03e4T + 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

−11.83343391490364981000777639055, −10.84726101585602912697721895784, −9.271989989653898454165823910815, −8.529685605599663516265393905571, −7.48700538365522744423094121686, −6.10877602407146541108154319102, −5.05307791406769457207849168895, −3.61824299976399470164633089295, −1.46994270815991637902921007129, −0.62445258124539311716451783425, 2.41655969477833661649222774965, 3.42385713808556408189656861097, 4.31752645131620556006845878574, 6.58840893187143643147768595301, 7.29185791633888445297380321127, 8.233168868560332888258640586218, 9.716120420431017603440657761599, 10.85194713320570252959714424415, 11.47909389830708197241721965089, 12.34338230244581497594595244782

Graph of the ZZ-function along the critical line