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Properties

Label 2-84-12.11-c5-0-27
Degree 22
Conductor 8484
Sign 0.757+0.652i0.757 + 0.652i
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
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  + (−1.03 − 5.56i)2-s + (13.7 − 7.34i)3-s + (−29.8 + 11.5i)4-s + 59.1i·5-s + (−55.1 − 68.8i)6-s + 49i·7-s + (95.2 + 153. i)8-s + (135. − 201. i)9-s + (329. − 61.4i)10-s + 242.·11-s + (−325. + 377. i)12-s + 886.·13-s + (272. − 50.9i)14-s + (434. + 813. i)15-s + (757. − 689. i)16-s − 988. i·17-s + ⋯
L(s)  = 1  + (−0.183 − 0.982i)2-s + (0.882 − 0.470i)3-s + (−0.932 + 0.361i)4-s + 1.05i·5-s + (−0.624 − 0.780i)6-s + 0.377i·7-s + (0.526 + 0.850i)8-s + (0.556 − 0.830i)9-s + (1.04 − 0.194i)10-s + 0.603·11-s + (−0.652 + 0.757i)12-s + 1.45·13-s + (0.371 − 0.0694i)14-s + (0.498 + 0.933i)15-s + (0.739 − 0.673i)16-s − 0.829i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.757+0.652i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((0.757+0.652i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.757+0.652i0.757 + 0.652i
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ84(71,⋅)\chi_{84} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), 0.757+0.652i)(2,\ 84,\ (\ :5/2),\ 0.757 + 0.652i)

Particular Values

L(3)L(3) ≈\approx 2.04326−0.758736i2.04326 - 0.758736i
L(12)L(\frac12) ≈\approx 2.04326−0.758736i2.04326 - 0.758736i
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)
bad2 1+(1.03+5.56i)T 1 + (1.03 + 5.56i)T
3 1+(−13.7+7.34i)T 1 + (-13.7 + 7.34i)T
7 1−49iT 1 - 49iT
good5 1−59.1iT−3.12e3T2 1 - 59.1iT - 3.12e3T^{2}
11 1−242.T+1.61e5T2 1 - 242.T + 1.61e5T^{2}
13 1−886.T+3.71e5T2 1 - 886.T + 3.71e5T^{2}
17 1+988.iT−1.41e6T2 1 + 988. iT - 1.41e6T^{2}
19 1−2.55e3iT−2.47e6T2 1 - 2.55e3iT - 2.47e6T^{2}
23 1−27.1T+6.43e6T2 1 - 27.1T + 6.43e6T^{2}
29 1−924.iT−2.05e7T2 1 - 924. iT - 2.05e7T^{2}
31 1+30.5iT−2.86e7T2 1 + 30.5iT - 2.86e7T^{2}
37 1−1.32e4T+6.93e7T2 1 - 1.32e4T + 6.93e7T^{2}
41 1+9.49e3iT−1.15e8T2 1 + 9.49e3iT - 1.15e8T^{2}
43 1−9.13e3iT−1.47e8T2 1 - 9.13e3iT - 1.47e8T^{2}
47 1+1.72e4T+2.29e8T2 1 + 1.72e4T + 2.29e8T^{2}
53 1−1.78e4iT−4.18e8T2 1 - 1.78e4iT - 4.18e8T^{2}
59 1+3.89e4T+7.14e8T2 1 + 3.89e4T + 7.14e8T^{2}
61 1−3.28e4T+8.44e8T2 1 - 3.28e4T + 8.44e8T^{2}
67 1−3.60e4iT−1.35e9T2 1 - 3.60e4iT - 1.35e9T^{2}
71 1−5.83e4T+1.80e9T2 1 - 5.83e4T + 1.80e9T^{2}
73 1+7.38e4T+2.07e9T2 1 + 7.38e4T + 2.07e9T^{2}
79 1+6.55e4iT−3.07e9T2 1 + 6.55e4iT - 3.07e9T^{2}
83 1−6.12e4T+3.93e9T2 1 - 6.12e4T + 3.93e9T^{2}
89 1+4.18e4iT−5.58e9T2 1 + 4.18e4iT - 5.58e9T^{2}
97 1−3.85e4T+8.58e9T2 1 - 3.85e4T + 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

−13.17717835805156145834132865113, −12.03937462073189799421523727935, −11.04608776152374624747532746809, −9.852775071929327037449034427960, −8.812691115393577047831962082557, −7.73322744416188930813446350074, −6.24179554323922921464737066071, −3.85397254147007287610688290261, −2.86296070811001385093946477186, −1.41064011626807765453477615904, 1.12379708109284480632934083649, 3.85563289441203898092407224313, 4.86806188023119784974727384152, 6.49041211748792226065644712391, 8.029765170527643402940988055429, 8.784268576746373102201662034261, 9.556877196476095240609404198066, 10.97375251117188018605576354232, 13.02259210184747444295499582885, 13.46355663816515382533884745715

Graph of the ZZ-function along the critical line