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Properties

Label 2-84-12.11-c3-0-28
Degree 22
Conductor 8484
Sign 0.439+0.898i0.439 + 0.898i
Analytic cond. 4.956164.95616
Root an. cond. 2.226242.22624
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.16 − 1.81i)2-s + (3.06 + 4.19i)3-s + (1.40 − 7.87i)4-s − 20.4i·5-s + (14.2 + 3.52i)6-s + 7i·7-s + (−11.2 − 19.6i)8-s + (−8.20 + 25.7i)9-s + (−37.1 − 44.3i)10-s + 32.3·11-s + (37.3 − 18.2i)12-s + 24.7·13-s + (12.7 + 15.1i)14-s + (85.8 − 62.7i)15-s + (−60.0 − 22.0i)16-s + 103. i·17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.590 + 0.807i)3-s + (0.175 − 0.984i)4-s − 1.83i·5-s + (0.970 + 0.240i)6-s + 0.377i·7-s + (−0.497 − 0.867i)8-s + (−0.303 + 0.952i)9-s + (−1.17 − 1.40i)10-s + 0.887·11-s + (0.898 − 0.439i)12-s + 0.527·13-s + (0.242 + 0.289i)14-s + (1.47 − 1.08i)15-s + (−0.938 − 0.344i)16-s + 1.48i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.439+0.898i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+3/2) L(s)=((0.439+0.898i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.439+0.898i0.439 + 0.898i
Analytic conductor: 4.956164.95616
Root analytic conductor: 2.226242.22624
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ84(71,⋅)\chi_{84} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :3/2), 0.439+0.898i)(2,\ 84,\ (\ :3/2),\ 0.439 + 0.898i)

Particular Values

L(2)L(2) ≈\approx 2.23883−1.39713i2.23883 - 1.39713i
L(12)L(\frac12) ≈\approx 2.23883−1.39713i2.23883 - 1.39713i
L(52)L(\frac{5}{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+(−2.16+1.81i)T 1 + (-2.16 + 1.81i)T
3 1+(−3.06−4.19i)T 1 + (-3.06 - 4.19i)T
7 1−7iT 1 - 7iT
good5 1+20.4iT−125T2 1 + 20.4iT - 125T^{2}
11 1−32.3T+1.33e3T2 1 - 32.3T + 1.33e3T^{2}
13 1−24.7T+2.19e3T2 1 - 24.7T + 2.19e3T^{2}
17 1−103.iT−4.91e3T2 1 - 103. iT - 4.91e3T^{2}
19 1−3.49iT−6.85e3T2 1 - 3.49iT - 6.85e3T^{2}
23 1−18.3T+1.21e4T2 1 - 18.3T + 1.21e4T^{2}
29 1+146.iT−2.43e4T2 1 + 146. iT - 2.43e4T^{2}
31 1−326.iT−2.97e4T2 1 - 326. iT - 2.97e4T^{2}
37 1−37.6T+5.06e4T2 1 - 37.6T + 5.06e4T^{2}
41 1−111.iT−6.89e4T2 1 - 111. iT - 6.89e4T^{2}
43 1+119.iT−7.95e4T2 1 + 119. iT - 7.95e4T^{2}
47 1−145.T+1.03e5T2 1 - 145.T + 1.03e5T^{2}
53 1+386.iT−1.48e5T2 1 + 386. iT - 1.48e5T^{2}
59 1−108.T+2.05e5T2 1 - 108.T + 2.05e5T^{2}
61 1+305.T+2.26e5T2 1 + 305.T + 2.26e5T^{2}
67 1−508.iT−3.00e5T2 1 - 508. iT - 3.00e5T^{2}
71 1−99.6T+3.57e5T2 1 - 99.6T + 3.57e5T^{2}
73 1+478.T+3.89e5T2 1 + 478.T + 3.89e5T^{2}
79 1+887.iT−4.93e5T2 1 + 887. iT - 4.93e5T^{2}
83 1−26.5T+5.71e5T2 1 - 26.5T + 5.71e5T^{2}
89 1+807.iT−7.04e5T2 1 + 807. iT - 7.04e5T^{2}
97 1−1.29e3T+9.12e5T2 1 - 1.29e3T + 9.12e5T^{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

−13.44480887679980154193289979563, −12.61920265279812531379522530719, −11.64108456147785031488594031772, −10.25630120297532286640451612346, −9.115899314668314946511547250674, −8.474346461177702836747746398756, −5.88476341596555632707010313845, −4.72026862935323924448425834079, −3.75717471232301240538865223635, −1.57716272302113645878034162113, 2.65553245235578808629773357056, 3.75635938897920182613074682828, 6.15326008540191995180315581846, 6.97580189231654458153925882662, 7.67615938197768359822455421518, 9.301540638230577911005136297101, 11.09021022851261240745937372107, 11.90174347636782443341628088639, 13.39452366176522346680257983742, 14.07524958977226603991603562014

Graph of the ZZ-function along the critical line