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Properties

Label 2-84-12.11-c5-0-0
Degree 22
Conductor 8484
Sign −0.000383+0.999i-0.000383 + 0.999i
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  + (1.70 + 5.39i)2-s + (−12.7 − 8.95i)3-s + (−26.2 + 18.3i)4-s + 95.2i·5-s + (26.5 − 84.0i)6-s − 49i·7-s + (−143. − 110. i)8-s + (82.6 + 228. i)9-s + (−513. + 162. i)10-s − 125.·11-s + (498. + 0.191i)12-s − 253.·13-s + (264. − 83.4i)14-s + (852. − 1.21e3i)15-s + (349. − 962. i)16-s − 1.11e3i·17-s + ⋯
L(s)  = 1  + (0.301 + 0.953i)2-s + (−0.818 − 0.574i)3-s + (−0.818 + 0.574i)4-s + 1.70i·5-s + (0.301 − 0.953i)6-s − 0.377i·7-s + (−0.793 − 0.608i)8-s + (0.340 + 0.940i)9-s + (−1.62 + 0.512i)10-s − 0.313·11-s + (0.999 + 0.000383i)12-s − 0.416·13-s + (0.360 − 0.113i)14-s + (0.978 − 1.39i)15-s + (0.340 − 0.940i)16-s − 0.934i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((−0.000383+0.999i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000383 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((−0.000383+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.000383 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −0.000383+0.999i-0.000383 + 0.999i
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), −0.000383+0.999i)(2,\ 84,\ (\ :5/2),\ -0.000383 + 0.999i)

Particular Values

L(3)L(3) ≈\approx 0.00723078−0.00723356i0.00723078 - 0.00723356i
L(12)L(\frac12) ≈\approx 0.00723078−0.00723356i0.00723078 - 0.00723356i
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+(−1.70−5.39i)T 1 + (-1.70 - 5.39i)T
3 1+(12.7+8.95i)T 1 + (12.7 + 8.95i)T
7 1+49iT 1 + 49iT
good5 1−95.2iT−3.12e3T2 1 - 95.2iT - 3.12e3T^{2}
11 1+125.T+1.61e5T2 1 + 125.T + 1.61e5T^{2}
13 1+253.T+3.71e5T2 1 + 253.T + 3.71e5T^{2}
17 1+1.11e3iT−1.41e6T2 1 + 1.11e3iT - 1.41e6T^{2}
19 1+1.55e3iT−2.47e6T2 1 + 1.55e3iT - 2.47e6T^{2}
23 1+237.T+6.43e6T2 1 + 237.T + 6.43e6T^{2}
29 1+7.32e3iT−2.05e7T2 1 + 7.32e3iT - 2.05e7T^{2}
31 1−7.61e3iT−2.86e7T2 1 - 7.61e3iT - 2.86e7T^{2}
37 1+2.78e3T+6.93e7T2 1 + 2.78e3T + 6.93e7T^{2}
41 1−1.81e4iT−1.15e8T2 1 - 1.81e4iT - 1.15e8T^{2}
43 1+6.77e3iT−1.47e8T2 1 + 6.77e3iT - 1.47e8T^{2}
47 1+1.99e4T+2.29e8T2 1 + 1.99e4T + 2.29e8T^{2}
53 1−1.89e4iT−4.18e8T2 1 - 1.89e4iT - 4.18e8T^{2}
59 1+4.32e4T+7.14e8T2 1 + 4.32e4T + 7.14e8T^{2}
61 1−1.23e4T+8.44e8T2 1 - 1.23e4T + 8.44e8T^{2}
67 1+4.92e4iT−1.35e9T2 1 + 4.92e4iT - 1.35e9T^{2}
71 1−3.10e4T+1.80e9T2 1 - 3.10e4T + 1.80e9T^{2}
73 1+4.59e4T+2.07e9T2 1 + 4.59e4T + 2.07e9T^{2}
79 1+8.17e4iT−3.07e9T2 1 + 8.17e4iT - 3.07e9T^{2}
83 1+5.75e4T+3.93e9T2 1 + 5.75e4T + 3.93e9T^{2}
89 1−1.20e5iT−5.58e9T2 1 - 1.20e5iT - 5.58e9T^{2}
97 1−2.52e4T+8.58e9T2 1 - 2.52e4T + 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

−14.03790671868331933255310383687, −13.38659272703216485445499013470, −11.97351508593036836629400822396, −10.96310339485530394306753575760, −9.819511335021466091694135506572, −7.78857630081930086528600037356, −7.01394550726800607019692172967, −6.26623653535599467586553638564, −4.80318401987669138133816759708, −2.88963713775046128894864088308, 0.00439549853971399165938518163, 1.50327736251664077892305301446, 3.91005136705617839747148647531, 5.01690161788142947800063626378, 5.79280058389726227413694622500, 8.403881803960862713021142643491, 9.385977383994689167093637818319, 10.33517335300376506811932269648, 11.60226593514410812409214860360, 12.50070633752654277509273000995

Graph of the ZZ-function along the critical line