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Properties

Label 2-3e4-81.7-c1-0-6
Degree 22
Conductor 8181
Sign 0.905+0.424i0.905 + 0.424i
Analytic cond. 0.6467880.646788
Root an. cond. 0.8042310.804231
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.470 + 0.631i)2-s + (−1.10 − 1.32i)3-s + (0.395 − 1.32i)4-s + (2.32 − 1.53i)5-s + (0.318 − 1.32i)6-s + (−3.41 + 3.62i)7-s + (2.50 − 0.910i)8-s + (−0.537 + 2.95i)9-s + (2.06 + 0.750i)10-s + (1.55 − 0.779i)11-s + (−2.19 + 0.940i)12-s + (0.859 + 1.99i)13-s + (−3.89 − 0.455i)14-s + (−4.61 − 1.39i)15-s + (−0.554 − 0.364i)16-s + (−3.39 + 2.84i)17-s + ⋯
L(s)  = 1  + (0.332 + 0.446i)2-s + (−0.640 − 0.767i)3-s + (0.197 − 0.660i)4-s + (1.04 − 0.684i)5-s + (0.129 − 0.541i)6-s + (−1.29 + 1.36i)7-s + (0.884 − 0.321i)8-s + (−0.179 + 0.983i)9-s + (0.651 + 0.237i)10-s + (0.468 − 0.235i)11-s + (−0.634 + 0.271i)12-s + (0.238 + 0.552i)13-s + (−1.04 − 0.121i)14-s + (−1.19 − 0.360i)15-s + (−0.138 − 0.0911i)16-s + (−0.823 + 0.690i)17-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.905+0.424i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+1/2) L(s)=((0.905+0.424i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.905+0.424i0.905 + 0.424i
Analytic conductor: 0.6467880.646788
Root analytic conductor: 0.8042310.804231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ81(7,⋅)\chi_{81} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 81, ( :1/2), 0.905+0.424i)(2,\ 81,\ (\ :1/2),\ 0.905 + 0.424i)

Particular Values

L(1)L(1) ≈\approx 1.00987−0.224772i1.00987 - 0.224772i
L(12)L(\frac12) ≈\approx 1.00987−0.224772i1.00987 - 0.224772i
L(32)L(\frac{3}{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)
bad3 1+(1.10+1.32i)T 1 + (1.10 + 1.32i)T
good2 1+(−0.470−0.631i)T+(−0.573+1.91i)T2 1 + (-0.470 - 0.631i)T + (-0.573 + 1.91i)T^{2}
5 1+(−2.32+1.53i)T+(1.98−4.59i)T2 1 + (-2.32 + 1.53i)T + (1.98 - 4.59i)T^{2}
7 1+(3.41−3.62i)T+(−0.407−6.98i)T2 1 + (3.41 - 3.62i)T + (-0.407 - 6.98i)T^{2}
11 1+(−1.55+0.779i)T+(6.56−8.82i)T2 1 + (-1.55 + 0.779i)T + (6.56 - 8.82i)T^{2}
13 1+(−0.859−1.99i)T+(−8.92+9.45i)T2 1 + (-0.859 - 1.99i)T + (-8.92 + 9.45i)T^{2}
17 1+(3.39−2.84i)T+(2.95−16.7i)T2 1 + (3.39 - 2.84i)T + (2.95 - 16.7i)T^{2}
19 1+(1.63+1.37i)T+(3.29+18.7i)T2 1 + (1.63 + 1.37i)T + (3.29 + 18.7i)T^{2}
23 1+(−0.465−0.493i)T+(−1.33+22.9i)T2 1 + (-0.465 - 0.493i)T + (-1.33 + 22.9i)T^{2}
29 1+(−5.71+0.668i)T+(28.2−6.68i)T2 1 + (-5.71 + 0.668i)T + (28.2 - 6.68i)T^{2}
31 1+(5.72−1.35i)T+(27.7−13.9i)T2 1 + (5.72 - 1.35i)T + (27.7 - 13.9i)T^{2}
37 1+(−0.131−0.747i)T+(−34.7+12.6i)T2 1 + (-0.131 - 0.747i)T + (-34.7 + 12.6i)T^{2}
41 1+(−0.0737+0.0990i)T+(−11.7−39.2i)T2 1 + (-0.0737 + 0.0990i)T + (-11.7 - 39.2i)T^{2}
43 1+(0.148−2.54i)T+(−42.7−4.99i)T2 1 + (0.148 - 2.54i)T + (-42.7 - 4.99i)T^{2}
47 1+(−1.65−0.391i)T+(42.0+21.0i)T2 1 + (-1.65 - 0.391i)T + (42.0 + 21.0i)T^{2}
53 1+(5.02+8.69i)T+(−26.5+45.8i)T2 1 + (5.02 + 8.69i)T + (-26.5 + 45.8i)T^{2}
59 1+(10.3+5.21i)T+(35.2+47.3i)T2 1 + (10.3 + 5.21i)T + (35.2 + 47.3i)T^{2}
61 1+(2.27+7.58i)T+(−50.9+33.5i)T2 1 + (2.27 + 7.58i)T + (-50.9 + 33.5i)T^{2}
67 1+(−0.462−0.0540i)T+(65.1+15.4i)T2 1 + (-0.462 - 0.0540i)T + (65.1 + 15.4i)T^{2}
71 1+(−11.4−4.17i)T+(54.3+45.6i)T2 1 + (-11.4 - 4.17i)T + (54.3 + 45.6i)T^{2}
73 1+(2.01−0.732i)T+(55.9−46.9i)T2 1 + (2.01 - 0.732i)T + (55.9 - 46.9i)T^{2}
79 1+(−4.18−5.62i)T+(−22.6+75.6i)T2 1 + (-4.18 - 5.62i)T + (-22.6 + 75.6i)T^{2}
83 1+(1.97+2.65i)T+(−23.8+79.5i)T2 1 + (1.97 + 2.65i)T + (-23.8 + 79.5i)T^{2}
89 1+(4.72−1.71i)T+(68.1−57.2i)T2 1 + (4.72 - 1.71i)T + (68.1 - 57.2i)T^{2}
97 1+(−6.27−4.12i)T+(38.4+89.0i)T2 1 + (-6.27 - 4.12i)T + (38.4 + 89.0i)T^{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

−14.08490272624711029381037393186, −13.14176340449461180238428012468, −12.51696827023224799632617551314, −11.14900048331360980102139326239, −9.732673311041363893839305098217, −8.780739740836086362857167110068, −6.53118194856908351262457286282, −6.19117784704273531809153657737, −5.12758723213643698044715777103, −1.96162584811106564359673526296, 3.09878609196109257559462121005, 4.28971000591971014807319959503, 6.23413646773448018757056195562, 7.10657487481799078710613704057, 9.324518027983059370893149376167, 10.38288920989507422787735279449, 10.86886455369434261241571860001, 12.33233462941760843870831494148, 13.33717688149255042428254495813, 14.12820440540439237880256385490

Graph of the ZZ-function along the critical line