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Properties

Label 2-147-21.20-c5-0-60
Degree 22
Conductor 147147
Sign 0.332−0.943i0.332 - 0.943i
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  − 4.29i·2-s + (−1.28 − 15.5i)3-s + 13.5·4-s − 75.7·5-s + (−66.6 + 5.52i)6-s − 195. i·8-s + (−239. + 40.0i)9-s + 324. i·10-s − 683. i·11-s + (−17.5 − 211. i)12-s + 904. i·13-s + (97.5 + 1.17e3i)15-s − 404.·16-s − 831.·17-s + (171. + 1.02e3i)18-s + 46.6i·19-s + ⋯
L(s)  = 1  − 0.758i·2-s + (−0.0825 − 0.996i)3-s + 0.424·4-s − 1.35·5-s + (−0.755 + 0.0626i)6-s − 1.08i·8-s + (−0.986 + 0.164i)9-s + 1.02i·10-s − 1.70i·11-s + (−0.0350 − 0.423i)12-s + 1.48i·13-s + (0.111 + 1.34i)15-s − 0.394·16-s − 0.697·17-s + (0.124 + 0.748i)18-s + 0.0296i·19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.332−0.943i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+5/2) L(s)=((0.332−0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 0.332−0.943i0.332 - 0.943i
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), 0.332−0.943i)(2,\ 147,\ (\ :5/2),\ 0.332 - 0.943i)

Particular Values

L(3)L(3) ≈\approx 0.18699359260.1869935926
L(12)L(\frac12) ≈\approx 0.18699359260.1869935926
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+(1.28+15.5i)T 1 + (1.28 + 15.5i)T
7 1 1
good2 1+4.29iT−32T2 1 + 4.29iT - 32T^{2}
5 1+75.7T+3.12e3T2 1 + 75.7T + 3.12e3T^{2}
11 1+683.iT−1.61e5T2 1 + 683. iT - 1.61e5T^{2}
13 1−904.iT−3.71e5T2 1 - 904. iT - 3.71e5T^{2}
17 1+831.T+1.41e6T2 1 + 831.T + 1.41e6T^{2}
19 1−46.6iT−2.47e6T2 1 - 46.6iT - 2.47e6T^{2}
23 1−3.22e3iT−6.43e6T2 1 - 3.22e3iT - 6.43e6T^{2}
29 1+644.iT−2.05e7T2 1 + 644. iT - 2.05e7T^{2}
31 1+818.iT−2.86e7T2 1 + 818. iT - 2.86e7T^{2}
37 1−1.25e4T+6.93e7T2 1 - 1.25e4T + 6.93e7T^{2}
41 1+3.58e3T+1.15e8T2 1 + 3.58e3T + 1.15e8T^{2}
43 1+1.27e4T+1.47e8T2 1 + 1.27e4T + 1.47e8T^{2}
47 1−5.08e3T+2.29e8T2 1 - 5.08e3T + 2.29e8T^{2}
53 1−2.92e4iT−4.18e8T2 1 - 2.92e4iT - 4.18e8T^{2}
59 1+1.42e4T+7.14e8T2 1 + 1.42e4T + 7.14e8T^{2}
61 1+1.02e3iT−8.44e8T2 1 + 1.02e3iT - 8.44e8T^{2}
67 1+6.56e3T+1.35e9T2 1 + 6.56e3T + 1.35e9T^{2}
71 1−2.16e4iT−1.80e9T2 1 - 2.16e4iT - 1.80e9T^{2}
73 1+5.61e4iT−2.07e9T2 1 + 5.61e4iT - 2.07e9T^{2}
79 1+2.45e4T+3.07e9T2 1 + 2.45e4T + 3.07e9T^{2}
83 1+1.13e5T+3.93e9T2 1 + 1.13e5T + 3.93e9T^{2}
89 1+1.20e4T+5.58e9T2 1 + 1.20e4T + 5.58e9T^{2}
97 1+1.31e5iT−8.58e9T2 1 + 1.31e5iT - 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

−11.51036109572201462064840964177, −10.98701510542629171588844095988, −9.129807670085293898887900189984, −8.036814070211465320338010909702, −7.07544834046876899528250199749, −6.08586205381863178758709018230, −4.00982292116556005436739873174, −2.89831289878840979204456459288, −1.36846938214234819755346834524, −0.06418457287978335987803504623, 2.72255658329901886893197552849, 4.22648151257293643908636613803, 5.16174548060585069648976691342, 6.65523804492098805017182470238, 7.73880294262258732320906075180, 8.489989433466123219265975043360, 10.01063597528495768105971247657, 10.91834697365677272769439191879, 11.80811555343756893292329577271, 12.74869750486089513589539142314

Graph of the ZZ-function along the critical line