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Properties

Label 2-147-21.20-c5-0-13
Degree 22
Conductor 147147
Sign −0.573+0.818i-0.573 + 0.818i
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  + 8.31i·2-s + (14.0 + 6.83i)3-s − 37.1·4-s − 28.6·5-s + (−56.8 + 116. i)6-s − 42.3i·8-s + (149. + 191. i)9-s − 238. i·10-s + 596. i·11-s + (−519. − 253. i)12-s + 602. i·13-s + (−401. − 196. i)15-s − 834.·16-s + 154.·17-s + (−1.59e3 + 1.24e3i)18-s − 2.86e3i·19-s + ⋯
L(s)  = 1  + 1.46i·2-s + (0.898 + 0.438i)3-s − 1.15·4-s − 0.513·5-s + (−0.644 + 1.32i)6-s − 0.234i·8-s + (0.615 + 0.788i)9-s − 0.754i·10-s + 1.48i·11-s + (−1.04 − 0.508i)12-s + 0.988i·13-s + (−0.461 − 0.225i)15-s − 0.815·16-s + 0.129·17-s + (−1.15 + 0.904i)18-s − 1.82i·19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((−0.573+0.818i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.818i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+5/2) L(s)=((−0.573+0.818i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: −0.573+0.818i-0.573 + 0.818i
Analytic conductor: 23.576423.5764
Root analytic conductor: 4.855554.85555
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ147(146,⋅)\chi_{147} (146, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :5/2), −0.573+0.818i)(2,\ 147,\ (\ :5/2),\ -0.573 + 0.818i)

Particular Values

L(3)L(3) ≈\approx 1.7379660141.737966014
L(12)L(\frac12) ≈\approx 1.7379660141.737966014
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+(−14.0−6.83i)T 1 + (-14.0 - 6.83i)T
7 1 1
good2 1−8.31iT−32T2 1 - 8.31iT - 32T^{2}
5 1+28.6T+3.12e3T2 1 + 28.6T + 3.12e3T^{2}
11 1−596.iT−1.61e5T2 1 - 596. iT - 1.61e5T^{2}
13 1−602.iT−3.71e5T2 1 - 602. iT - 3.71e5T^{2}
17 1−154.T+1.41e6T2 1 - 154.T + 1.41e6T^{2}
19 1+2.86e3iT−2.47e6T2 1 + 2.86e3iT - 2.47e6T^{2}
23 1+3.78e3iT−6.43e6T2 1 + 3.78e3iT - 6.43e6T^{2}
29 1−2.85e3iT−2.05e7T2 1 - 2.85e3iT - 2.05e7T^{2}
31 1−3.22e3iT−2.86e7T2 1 - 3.22e3iT - 2.86e7T^{2}
37 1+1.04e4T+6.93e7T2 1 + 1.04e4T + 6.93e7T^{2}
41 1+6.92e3T+1.15e8T2 1 + 6.92e3T + 1.15e8T^{2}
43 1−3.80e3T+1.47e8T2 1 - 3.80e3T + 1.47e8T^{2}
47 1−1.57e4T+2.29e8T2 1 - 1.57e4T + 2.29e8T^{2}
53 1−4.78e3iT−4.18e8T2 1 - 4.78e3iT - 4.18e8T^{2}
59 1−3.34e4T+7.14e8T2 1 - 3.34e4T + 7.14e8T^{2}
61 1−4.51e4iT−8.44e8T2 1 - 4.51e4iT - 8.44e8T^{2}
67 1+4.95e4T+1.35e9T2 1 + 4.95e4T + 1.35e9T^{2}
71 1−3.59e4iT−1.80e9T2 1 - 3.59e4iT - 1.80e9T^{2}
73 1−2.67e4iT−2.07e9T2 1 - 2.67e4iT - 2.07e9T^{2}
79 1−6.05e3T+3.07e9T2 1 - 6.05e3T + 3.07e9T^{2}
83 1−3.62e4T+3.93e9T2 1 - 3.62e4T + 3.93e9T^{2}
89 1−7.63e4T+5.58e9T2 1 - 7.63e4T + 5.58e9T^{2}
97 1−5.19e4iT−8.58e9T2 1 - 5.19e4iT - 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

−13.22815590550863334210579554305, −11.93373084893535618200888348762, −10.48437881883470860905538944222, −9.194827982035599465556132803089, −8.579472992510139660807483140883, −7.30320075550936058803405252217, −6.87532680747972025371900572654, −4.95280336983184359835980275803, −4.24052745381365302478500142369, −2.31303060950545163854569927782, 0.51574989127411784404867736323, 1.78533236293613374808074709750, 3.30340951809081792423138486185, 3.72017855290844555337991239136, 5.86379253329199021023252385137, 7.64756277310057608220927351930, 8.435075542801821340883782424189, 9.619223716011959293621594174023, 10.52516429873967837019093553481, 11.65188339382437985972401278205

Graph of the ZZ-function along the critical line