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Properties

Label 2-84-7.4-c5-0-6
Degree 22
Conductor 8484
Sign −0.553+0.832i-0.553 + 0.832i
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  + (4.5 − 7.79i)3-s + (−23.0 − 39.9i)5-s + (112. + 64.8i)7-s + (−40.5 − 70.1i)9-s + (315. − 546. i)11-s − 1.07e3·13-s − 415.·15-s + (−80.5 + 139. i)17-s + (−588. − 1.01e3i)19-s + (1.01e3 − 583. i)21-s + (−1.08e3 − 1.87e3i)23-s + (499. − 864. i)25-s − 729·27-s − 4.49e3·29-s + (−159. + 275. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.412 − 0.714i)5-s + (0.866 + 0.500i)7-s + (−0.166 − 0.288i)9-s + (0.786 − 1.36i)11-s − 1.77·13-s − 0.476·15-s + (−0.0676 + 0.117i)17-s + (−0.373 − 0.647i)19-s + (0.500 − 0.288i)21-s + (−0.426 − 0.738i)23-s + (0.159 − 0.276i)25-s − 0.192·27-s − 0.991·29-s + (−0.0297 + 0.0515i)31-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((−0.553+0.832i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((−0.553+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −0.553+0.832i-0.553 + 0.832i
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ84(25,⋅)\chi_{84} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), −0.553+0.832i)(2,\ 84,\ (\ :5/2),\ -0.553 + 0.832i)

Particular Values

L(3)L(3) ≈\approx 0.709868−1.32480i0.709868 - 1.32480i
L(12)L(\frac12) ≈\approx 0.709868−1.32480i0.709868 - 1.32480i
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
3 1+(−4.5+7.79i)T 1 + (-4.5 + 7.79i)T
7 1+(−112.−64.8i)T 1 + (-112. - 64.8i)T
good5 1+(23.0+39.9i)T+(−1.56e3+2.70e3i)T2 1 + (23.0 + 39.9i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−315.+546.i)T+(−8.05e4−1.39e5i)T2 1 + (-315. + 546. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+1.07e3T+3.71e5T2 1 + 1.07e3T + 3.71e5T^{2}
17 1+(80.5−139.i)T+(−7.09e5−1.22e6i)T2 1 + (80.5 - 139. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(588.+1.01e3i)T+(−1.23e6+2.14e6i)T2 1 + (588. + 1.01e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(1.08e3+1.87e3i)T+(−3.21e6+5.57e6i)T2 1 + (1.08e3 + 1.87e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+4.49e3T+2.05e7T2 1 + 4.49e3T + 2.05e7T^{2}
31 1+(159.−275.i)T+(−1.43e7−2.47e7i)T2 1 + (159. - 275. i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(7.59e3+1.31e4i)T+(−3.46e7+6.00e7i)T2 1 + (7.59e3 + 1.31e4i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−2.05e4T+1.15e8T2 1 - 2.05e4T + 1.15e8T^{2}
43 1+455.T+1.47e8T2 1 + 455.T + 1.47e8T^{2}
47 1+(−1.03e4−1.79e4i)T+(−1.14e8+1.98e8i)T2 1 + (-1.03e4 - 1.79e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−9.65e3+1.67e4i)T+(−2.09e8−3.62e8i)T2 1 + (-9.65e3 + 1.67e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(3.18e3−5.51e3i)T+(−3.57e8−6.19e8i)T2 1 + (3.18e3 - 5.51e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−2.45e4−4.25e4i)T+(−4.22e8+7.31e8i)T2 1 + (-2.45e4 - 4.25e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(1.70e4−2.94e4i)T+(−6.75e8−1.16e9i)T2 1 + (1.70e4 - 2.94e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−6.29e4T+1.80e9T2 1 - 6.29e4T + 1.80e9T^{2}
73 1+(−4.43e3+7.67e3i)T+(−1.03e9−1.79e9i)T2 1 + (-4.43e3 + 7.67e3i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(1.72e4+2.98e4i)T+(−1.53e9+2.66e9i)T2 1 + (1.72e4 + 2.98e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+7.04e3T+3.93e9T2 1 + 7.04e3T + 3.93e9T^{2}
89 1+(−1.01e4−1.75e4i)T+(−2.79e9+4.83e9i)T2 1 + (-1.01e4 - 1.75e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−5.40e4T+8.58e9T2 1 - 5.40e4T + 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

−12.72890334177401793563532205446, −12.01495564874337591776574733795, −11.00711257016931531405157934910, −9.182087485484873618611260321011, −8.447653488393144275511144238186, −7.32389543320265269165928588100, −5.69917077417117844043269322620, −4.31797698229704536341713315901, −2.36035497088433317751405215993, −0.59708451868360033029805090933, 2.05863421924980949923692904651, 3.87601836887854677612210552990, 4.97142498976516659030371445022, 7.05698464223477029641445075135, 7.76275016492297043572347826177, 9.438423143405865236972338388396, 10.29674782473129323001954977895, 11.46719893314675671292244673145, 12.40306779388706635323732208957, 14.07675148872478654453447857927

Graph of the ZZ-function along the critical line