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Properties

Label 2-3e4-81.22-c1-0-1
Degree 22
Conductor 8181
Sign 0.0621−0.998i0.0621 - 0.998i
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  + (1.18 + 1.26i)2-s + (−1.21 + 1.23i)3-s + (−0.0583 + 1.00i)4-s + (−1.90 + 0.222i)5-s + (−3.00 − 0.0611i)6-s + (4.00 + 2.00i)7-s + (1.32 − 1.10i)8-s + (−0.0522 − 2.99i)9-s + (−2.54 − 2.13i)10-s + (−2.00 − 4.65i)11-s + (−1.16 − 1.28i)12-s + (−1.48 − 0.352i)13-s + (2.22 + 7.43i)14-s + (2.03 − 2.61i)15-s + (4.96 + 0.580i)16-s + (−0.703 + 3.99i)17-s + ⋯
L(s)  = 1  + (0.841 + 0.891i)2-s + (−0.700 + 0.713i)3-s + (−0.0291 + 0.501i)4-s + (−0.850 + 0.0994i)5-s + (−1.22 − 0.0249i)6-s + (1.51 + 0.759i)7-s + (0.467 − 0.392i)8-s + (−0.0174 − 0.999i)9-s + (−0.804 − 0.674i)10-s + (−0.605 − 1.40i)11-s + (−0.337 − 0.372i)12-s + (−0.412 − 0.0978i)13-s + (0.594 + 1.98i)14-s + (0.525 − 0.676i)15-s + (1.24 + 0.145i)16-s + (−0.170 + 0.967i)17-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.0621−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0621 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+1/2) L(s)=((0.0621−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0621 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.0621−0.998i0.0621 - 0.998i
Analytic conductor: 0.6467880.646788
Root analytic conductor: 0.8042310.804231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ81(22,⋅)\chi_{81} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 81, ( :1/2), 0.0621−0.998i)(2,\ 81,\ (\ :1/2),\ 0.0621 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.838874+0.788290i0.838874 + 0.788290i
L(12)L(\frac12) ≈\approx 0.838874+0.788290i0.838874 + 0.788290i
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+(1.21−1.23i)T 1 + (1.21 - 1.23i)T
good2 1+(−1.18−1.26i)T+(−0.116+1.99i)T2 1 + (-1.18 - 1.26i)T + (-0.116 + 1.99i)T^{2}
5 1+(1.90−0.222i)T+(4.86−1.15i)T2 1 + (1.90 - 0.222i)T + (4.86 - 1.15i)T^{2}
7 1+(−4.00−2.00i)T+(4.18+5.61i)T2 1 + (-4.00 - 2.00i)T + (4.18 + 5.61i)T^{2}
11 1+(2.00+4.65i)T+(−7.54+8.00i)T2 1 + (2.00 + 4.65i)T + (-7.54 + 8.00i)T^{2}
13 1+(1.48+0.352i)T+(11.6+5.83i)T2 1 + (1.48 + 0.352i)T + (11.6 + 5.83i)T^{2}
17 1+(0.703−3.99i)T+(−15.9−5.81i)T2 1 + (0.703 - 3.99i)T + (-15.9 - 5.81i)T^{2}
19 1+(0.430+2.44i)T+(−17.8+6.49i)T2 1 + (0.430 + 2.44i)T + (-17.8 + 6.49i)T^{2}
23 1+(1.51−0.761i)T+(13.7−18.4i)T2 1 + (1.51 - 0.761i)T + (13.7 - 18.4i)T^{2}
29 1+(0.835−2.79i)T+(−24.2−15.9i)T2 1 + (0.835 - 2.79i)T + (-24.2 - 15.9i)T^{2}
31 1+(4.90+3.22i)T+(12.2+28.4i)T2 1 + (4.90 + 3.22i)T + (12.2 + 28.4i)T^{2}
37 1+(3.32−1.20i)T+(28.3−23.7i)T2 1 + (3.32 - 1.20i)T + (28.3 - 23.7i)T^{2}
41 1+(−3.07+3.25i)T+(−2.38−40.9i)T2 1 + (-3.07 + 3.25i)T + (-2.38 - 40.9i)T^{2}
43 1+(−2.67+3.58i)T+(−12.3−41.1i)T2 1 + (-2.67 + 3.58i)T + (-12.3 - 41.1i)T^{2}
47 1+(−2.58+1.69i)T+(18.6−43.1i)T2 1 + (-2.58 + 1.69i)T + (18.6 - 43.1i)T^{2}
53 1+(1.49−2.59i)T+(−26.5−45.8i)T2 1 + (1.49 - 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(0.820−1.90i)T+(−40.4−42.9i)T2 1 + (0.820 - 1.90i)T + (-40.4 - 42.9i)T^{2}
61 1+(−0.490−8.41i)T+(−60.5+7.08i)T2 1 + (-0.490 - 8.41i)T + (-60.5 + 7.08i)T^{2}
67 1+(3.13+10.4i)T+(−55.9+36.8i)T2 1 + (3.13 + 10.4i)T + (-55.9 + 36.8i)T^{2}
71 1+(0.318+0.267i)T+(12.3+69.9i)T2 1 + (0.318 + 0.267i)T + (12.3 + 69.9i)T^{2}
73 1+(6.86−5.76i)T+(12.6−71.8i)T2 1 + (6.86 - 5.76i)T + (12.6 - 71.8i)T^{2}
79 1+(−10.7−11.4i)T+(−4.59+78.8i)T2 1 + (-10.7 - 11.4i)T + (-4.59 + 78.8i)T^{2}
83 1+(−5.35−5.68i)T+(−4.82+82.8i)T2 1 + (-5.35 - 5.68i)T + (-4.82 + 82.8i)T^{2}
89 1+(−2.88+2.42i)T+(15.4−87.6i)T2 1 + (-2.88 + 2.42i)T + (15.4 - 87.6i)T^{2}
97 1+(9.96+1.16i)T+(94.3+22.3i)T2 1 + (9.96 + 1.16i)T + (94.3 + 22.3i)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

−15.02028596861524577892615535035, −13.93130082997937089536379870021, −12.44529086975534872580373083373, −11.33873834974220217039139551898, −10.66868056506696101362494031101, −8.689657716205243292177990241729, −7.59186950639378333594761214665, −5.89283430320555612784352660314, −5.14578873505365701768439845558, −3.93079967434527822511745579353, 1.96312216619758705253530953510, 4.35696923393780539344019526561, 5.05040201720230738475175052119, 7.44174033130949592679796161174, 7.83299847141214451885489072234, 10.36381177766010410990416254433, 11.28903227997110831452976369759, 11.95358577183355471436931988310, 12.72562138713614816764132224673, 13.86676805152873698119437523414

Graph of the ZZ-function along the critical line