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Properties

Label 2-84-12.11-c5-0-21
Degree 22
Conductor 8484
Sign −0.0514−0.998i-0.0514 - 0.998i
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  + (−3.60 + 4.36i)2-s + (−2.15 + 15.4i)3-s + (−6.05 − 31.4i)4-s − 60.5i·5-s + (−59.5 − 65.0i)6-s + 49i·7-s + (158. + 86.7i)8-s + (−233. − 66.5i)9-s + (264. + 218. i)10-s + 697.·11-s + (498. − 25.6i)12-s + 189.·13-s + (−213. − 176. i)14-s + (934. + 130. i)15-s + (−950. + 380. i)16-s − 419. i·17-s + ⋯
L(s)  = 1  + (−0.636 + 0.771i)2-s + (−0.138 + 0.990i)3-s + (−0.189 − 0.981i)4-s − 1.08i·5-s + (−0.675 − 0.737i)6-s + 0.377i·7-s + (0.877 + 0.479i)8-s + (−0.961 − 0.274i)9-s + (0.835 + 0.689i)10-s + 1.73·11-s + (0.998 − 0.0514i)12-s + 0.311·13-s + (−0.291 − 0.240i)14-s + (1.07 + 0.149i)15-s + (−0.928 + 0.371i)16-s − 0.351i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((−0.0514−0.998i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0514 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((−0.0514−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0514 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −0.0514−0.998i-0.0514 - 0.998i
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), −0.0514−0.998i)(2,\ 84,\ (\ :5/2),\ -0.0514 - 0.998i)

Particular Values

L(3)L(3) ≈\approx 0.845969+0.890714i0.845969 + 0.890714i
L(12)L(\frac12) ≈\approx 0.845969+0.890714i0.845969 + 0.890714i
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+(3.60−4.36i)T 1 + (3.60 - 4.36i)T
3 1+(2.15−15.4i)T 1 + (2.15 - 15.4i)T
7 1−49iT 1 - 49iT
good5 1+60.5iT−3.12e3T2 1 + 60.5iT - 3.12e3T^{2}
11 1−697.T+1.61e5T2 1 - 697.T + 1.61e5T^{2}
13 1−189.T+3.71e5T2 1 - 189.T + 3.71e5T^{2}
17 1+419.iT−1.41e6T2 1 + 419. iT - 1.41e6T^{2}
19 1−1.89e3iT−2.47e6T2 1 - 1.89e3iT - 2.47e6T^{2}
23 1+589.T+6.43e6T2 1 + 589.T + 6.43e6T^{2}
29 1−1.91e3iT−2.05e7T2 1 - 1.91e3iT - 2.05e7T^{2}
31 1−6.67e3iT−2.86e7T2 1 - 6.67e3iT - 2.86e7T^{2}
37 1−1.14e4T+6.93e7T2 1 - 1.14e4T + 6.93e7T^{2}
41 1+717.iT−1.15e8T2 1 + 717. iT - 1.15e8T^{2}
43 1−7.80e3iT−1.47e8T2 1 - 7.80e3iT - 1.47e8T^{2}
47 1+1.70e3T+2.29e8T2 1 + 1.70e3T + 2.29e8T^{2}
53 1−2.54e4iT−4.18e8T2 1 - 2.54e4iT - 4.18e8T^{2}
59 1−4.74e4T+7.14e8T2 1 - 4.74e4T + 7.14e8T^{2}
61 1−2.52e4T+8.44e8T2 1 - 2.52e4T + 8.44e8T^{2}
67 1+5.30e4iT−1.35e9T2 1 + 5.30e4iT - 1.35e9T^{2}
71 1−3.37e4T+1.80e9T2 1 - 3.37e4T + 1.80e9T^{2}
73 1−5.88e4T+2.07e9T2 1 - 5.88e4T + 2.07e9T^{2}
79 1−7.38e4iT−3.07e9T2 1 - 7.38e4iT - 3.07e9T^{2}
83 1+7.27e4T+3.93e9T2 1 + 7.27e4T + 3.93e9T^{2}
89 1+1.01e5iT−5.58e9T2 1 + 1.01e5iT - 5.58e9T^{2}
97 1+1.46e5T+8.58e9T2 1 + 1.46e5T + 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

−14.08242395185165387756416883500, −12.33782017716751329390074724336, −11.23181537750882690261248062022, −9.816014030739595642020780582258, −9.077746373430728753138199170384, −8.319886394346975062194456434214, −6.41936903324274852523888893362, −5.30075737068808335343073004753, −4.09546841215764554983837162853, −1.14781117750863802795734198282, 0.835045688503849014219927891417, 2.35484528161638255040368045021, 3.83458813936068047064090385623, 6.45904629044014448308216835371, 7.20449891306249539948402336907, 8.515593300732691106529516100557, 9.772161608698479866559282812590, 11.22717696478656087588575022698, 11.52581820210158371791847854440, 12.89956298219521311708702349084

Graph of the ZZ-function along the critical line