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Properties

Label 2-3e4-81.22-c1-0-6
Degree 22
Conductor 8181
Sign 0.606+0.795i0.606 + 0.795i
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  + (0.259 + 0.275i)2-s + (−0.322 − 1.70i)3-s + (0.107 − 1.85i)4-s + (−1.27 + 0.148i)5-s + (0.384 − 0.530i)6-s + (1.70 + 0.857i)7-s + (1.11 − 0.938i)8-s + (−2.79 + 1.09i)9-s + (−0.371 − 0.312i)10-s + (1.93 + 4.48i)11-s + (−3.18 + 0.413i)12-s + (3.34 + 0.793i)13-s + (0.207 + 0.692i)14-s + (0.663 + 2.11i)15-s + (−3.13 − 0.366i)16-s + (0.0728 − 0.412i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.194i)2-s + (−0.185 − 0.982i)3-s + (0.0539 − 0.926i)4-s + (−0.569 + 0.0665i)5-s + (0.157 − 0.216i)6-s + (0.645 + 0.324i)7-s + (0.395 − 0.331i)8-s + (−0.930 + 0.365i)9-s + (−0.117 − 0.0986i)10-s + (0.582 + 1.35i)11-s + (−0.920 + 0.119i)12-s + (0.928 + 0.220i)13-s + (0.0554 + 0.185i)14-s + (0.171 + 0.547i)15-s + (−0.784 − 0.0917i)16-s + (0.0176 − 0.100i)17-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.606+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+1/2) L(s)=((0.606+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.606+0.795i0.606 + 0.795i
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), 0.606+0.795i)(2,\ 81,\ (\ :1/2),\ 0.606 + 0.795i)

Particular Values

L(1)L(1) ≈\approx 0.874812−0.433171i0.874812 - 0.433171i
L(12)L(\frac12) ≈\approx 0.874812−0.433171i0.874812 - 0.433171i
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+(0.322+1.70i)T 1 + (0.322 + 1.70i)T
good2 1+(−0.259−0.275i)T+(−0.116+1.99i)T2 1 + (-0.259 - 0.275i)T + (-0.116 + 1.99i)T^{2}
5 1+(1.27−0.148i)T+(4.86−1.15i)T2 1 + (1.27 - 0.148i)T + (4.86 - 1.15i)T^{2}
7 1+(−1.70−0.857i)T+(4.18+5.61i)T2 1 + (-1.70 - 0.857i)T + (4.18 + 5.61i)T^{2}
11 1+(−1.93−4.48i)T+(−7.54+8.00i)T2 1 + (-1.93 - 4.48i)T + (-7.54 + 8.00i)T^{2}
13 1+(−3.34−0.793i)T+(11.6+5.83i)T2 1 + (-3.34 - 0.793i)T + (11.6 + 5.83i)T^{2}
17 1+(−0.0728+0.412i)T+(−15.9−5.81i)T2 1 + (-0.0728 + 0.412i)T + (-15.9 - 5.81i)T^{2}
19 1+(0.626+3.55i)T+(−17.8+6.49i)T2 1 + (0.626 + 3.55i)T + (-17.8 + 6.49i)T^{2}
23 1+(5.99−3.01i)T+(13.7−18.4i)T2 1 + (5.99 - 3.01i)T + (13.7 - 18.4i)T^{2}
29 1+(−1.44+4.81i)T+(−24.2−15.9i)T2 1 + (-1.44 + 4.81i)T + (-24.2 - 15.9i)T^{2}
31 1+(−5.54−3.64i)T+(12.2+28.4i)T2 1 + (-5.54 - 3.64i)T + (12.2 + 28.4i)T^{2}
37 1+(−0.465+0.169i)T+(28.3−23.7i)T2 1 + (-0.465 + 0.169i)T + (28.3 - 23.7i)T^{2}
41 1+(−4.16+4.41i)T+(−2.38−40.9i)T2 1 + (-4.16 + 4.41i)T + (-2.38 - 40.9i)T^{2}
43 1+(6.92−9.30i)T+(−12.3−41.1i)T2 1 + (6.92 - 9.30i)T + (-12.3 - 41.1i)T^{2}
47 1+(5.06−3.33i)T+(18.6−43.1i)T2 1 + (5.06 - 3.33i)T + (18.6 - 43.1i)T^{2}
53 1+(−2.95+5.10i)T+(−26.5−45.8i)T2 1 + (-2.95 + 5.10i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2.23+5.17i)T+(−40.4−42.9i)T2 1 + (-2.23 + 5.17i)T + (-40.4 - 42.9i)T^{2}
61 1+(−0.161−2.77i)T+(−60.5+7.08i)T2 1 + (-0.161 - 2.77i)T + (-60.5 + 7.08i)T^{2}
67 1+(−2.07−6.93i)T+(−55.9+36.8i)T2 1 + (-2.07 - 6.93i)T + (-55.9 + 36.8i)T^{2}
71 1+(11.9+10.0i)T+(12.3+69.9i)T2 1 + (11.9 + 10.0i)T + (12.3 + 69.9i)T^{2}
73 1+(5.04−4.23i)T+(12.6−71.8i)T2 1 + (5.04 - 4.23i)T + (12.6 - 71.8i)T^{2}
79 1+(2.62+2.78i)T+(−4.59+78.8i)T2 1 + (2.62 + 2.78i)T + (-4.59 + 78.8i)T^{2}
83 1+(−3.32−3.52i)T+(−4.82+82.8i)T2 1 + (-3.32 - 3.52i)T + (-4.82 + 82.8i)T^{2}
89 1+(7.97−6.69i)T+(15.4−87.6i)T2 1 + (7.97 - 6.69i)T + (15.4 - 87.6i)T^{2}
97 1+(−2.14−0.251i)T+(94.3+22.3i)T2 1 + (-2.14 - 0.251i)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

−14.22215875135161313944160586624, −13.28006667059237293768355894649, −11.87835531060008224664027907127, −11.32074564646154471386920530588, −9.790742593875732048047470426518, −8.294546716258683424721739727282, −7.08168920326359930500471613984, −6.03512718996633684943517999056, −4.56187894543284082678708233688, −1.76092800498733008326588397997, 3.47355894831640250403254915817, 4.29511485383354791600224903577, 6.07136904630790737898561641989, 8.083861829413223929873640103018, 8.624975573484237862577835163050, 10.41559780063554598574297468133, 11.40001664117603995699720999852, 11.98250274988202838958745375086, 13.58267305045831086682175528865, 14.44229368476028323732606301151

Graph of the ZZ-function along the critical line