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Properties

Label 2-147-147.134-c0-0-0
Degree 22
Conductor 147147
Sign 0.801−0.598i0.801 - 0.598i
Analytic cond. 0.07336250.0733625
Root an. cond. 0.2708550.270855
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)12-s + (−0.277 − 1.21i)13-s + (0.623 − 0.781i)16-s − 0.445·19-s + (0.623 − 0.781i)21-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)27-s + (0.623 + 0.781i)28-s − 1.80·31-s + (−0.222 − 0.974i)36-s + (1.62 + 0.781i)37-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)12-s + (−0.277 − 1.21i)13-s + (0.623 − 0.781i)16-s − 0.445·19-s + (0.623 − 0.781i)21-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)27-s + (0.623 + 0.781i)28-s − 1.80·31-s + (−0.222 − 0.974i)36-s + (1.62 + 0.781i)37-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.801−0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s) L(s)=((0.801−0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 0.801−0.598i0.801 - 0.598i
Analytic conductor: 0.07336250.0733625
Root analytic conductor: 0.2708550.270855
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ147(134,⋅)\chi_{147} (134, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :0), 0.801−0.598i)(2,\ 147,\ (\ :0),\ 0.801 - 0.598i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62824946000.6282494600
L(12)L(\frac12) ≈\approx 0.62824946000.6282494600
L(1)L(1) 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+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
7 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
good2 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
5 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
11 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
13 1+(0.277+1.21i)T+(−0.900+0.433i)T2 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2}
17 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
19 1+0.445T+T2 1 + 0.445T + T^{2}
23 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
29 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
31 1+1.80T+T2 1 + 1.80T + T^{2}
37 1+(−1.62−0.781i)T+(0.623+0.781i)T2 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2}
41 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
43 1+(0.277−0.347i)T+(−0.222−0.974i)T2 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2}
47 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
53 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
59 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
61 1+(−1.62−0.781i)T+(0.623+0.781i)T2 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2}
67 1−1.24T+T2 1 - 1.24T + T^{2}
71 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
73 1+(−0.400+1.75i)T+(−0.900−0.433i)T2 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2}
79 1+1.80T+T2 1 + 1.80T + T^{2}
83 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
89 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
97 1+0.445T+T2 1 + 0.445T + T^{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

−13.29857992924145003477403868077, −12.88644052026460720973490967709, −11.14616323085050135116168379626, −10.13298575402331494128947906773, −9.399419790678040583871006831579, −8.245582394742496889386483699207, −7.43547382881228174632113469317, −5.37823019688536841080327108404, −4.19462357919782614420479415634, −3.21284140013096734124203552472, 2.17226243928963024610178804391, 4.02941450391131317545777145254, 5.62358814613490033814542175522, 6.74045206617909125982328451203, 8.202845077853546682508785480288, 9.056450249933553734618448309680, 9.716622130236253542420350773433, 11.39023318691897282337833823622, 12.51042067151936381790558136735, 13.12847629958155816232984445796

Graph of the ZZ-function along the critical line