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Properties

Label 2-3e4-27.11-c8-0-3
Degree 22
Conductor 8181
Sign 0.784+0.620i0.784 + 0.620i
Analytic cond. 32.997632.9976
Root an. cond. 5.744355.74435
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−14.0 + 16.7i)2-s + (−38.4 − 217. i)4-s + (−373. + 1.02e3i)5-s + (−91.6 + 519. i)7-s + (−659. − 380. i)8-s + (−1.19e4 − 2.06e4i)10-s + (2.92e3 + 8.03e3i)11-s + (−2.76e4 + 2.31e4i)13-s + (−7.40e3 − 8.83e3i)14-s + (6.88e4 − 2.50e4i)16-s + (−9.04e4 + 5.22e4i)17-s + (−5.02e4 + 8.71e4i)19-s + (2.37e5 + 4.18e4i)20-s + (−1.75e5 − 6.38e4i)22-s + (−3.58e4 + 6.32e3i)23-s + ⋯
L(s)  = 1  + (−0.877 + 1.04i)2-s + (−0.150 − 0.850i)4-s + (−0.596 + 1.64i)5-s + (−0.0381 + 0.216i)7-s + (−0.160 − 0.0929i)8-s + (−1.19 − 2.06i)10-s + (0.199 + 0.548i)11-s + (−0.966 + 0.811i)13-s + (−0.192 − 0.229i)14-s + (1.05 − 0.382i)16-s + (−1.08 + 0.625i)17-s + (−0.385 + 0.668i)19-s + (1.48 + 0.261i)20-s + (−0.748 − 0.272i)22-s + (−0.128 + 0.0226i)23-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.784+0.620i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+4) L(s)=((0.784+0.620i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.784+0.620i0.784 + 0.620i
Analytic conductor: 32.997632.9976
Root analytic conductor: 5.744355.74435
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ81(35,⋅)\chi_{81} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 81, ( :4), 0.784+0.620i)(2,\ 81,\ (\ :4),\ 0.784 + 0.620i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.337759−0.117378i0.337759 - 0.117378i
L(12)L(\frac12) ≈\approx 0.337759−0.117378i0.337759 - 0.117378i
L(5)L(5) 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
good2 1+(14.0−16.7i)T+(−44.4−252.i)T2 1 + (14.0 - 16.7i)T + (-44.4 - 252. i)T^{2}
5 1+(373.−1.02e3i)T+(−2.99e5−2.51e5i)T2 1 + (373. - 1.02e3i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(91.6−519.i)T+(−5.41e6−1.97e6i)T2 1 + (91.6 - 519. i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(−2.92e3−8.03e3i)T+(−1.64e8+1.37e8i)T2 1 + (-2.92e3 - 8.03e3i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(2.76e4−2.31e4i)T+(1.41e8−8.03e8i)T2 1 + (2.76e4 - 2.31e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(9.04e4−5.22e4i)T+(3.48e9−6.04e9i)T2 1 + (9.04e4 - 5.22e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(5.02e4−8.71e4i)T+(−8.49e9−1.47e10i)T2 1 + (5.02e4 - 8.71e4i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(3.58e4−6.32e3i)T+(7.35e10−2.67e10i)T2 1 + (3.58e4 - 6.32e3i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(6.37e5−7.60e5i)T+(−8.68e10−4.92e11i)T2 1 + (6.37e5 - 7.60e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(1.81e5+1.03e6i)T+(−8.01e11+2.91e11i)T2 1 + (1.81e5 + 1.03e6i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(−1.04e6−1.80e6i)T+(−1.75e12+3.04e12i)T2 1 + (-1.04e6 - 1.80e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(−5.27e5−6.28e5i)T+(−1.38e12+7.86e12i)T2 1 + (-5.27e5 - 6.28e5i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(−2.50e6+9.12e5i)T+(8.95e12−7.51e12i)T2 1 + (-2.50e6 + 9.12e5i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(−8.34e6−1.47e6i)T+(2.23e13+8.14e12i)T2 1 + (-8.34e6 - 1.47e6i)T + (2.23e13 + 8.14e12i)T^{2}
53 1−5.29e6iT−6.22e13T2 1 - 5.29e6iT - 6.22e13T^{2}
59 1+(−5.70e6+1.56e7i)T+(−1.12e14−9.43e13i)T2 1 + (-5.70e6 + 1.56e7i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(1.10e5−6.24e5i)T+(−1.80e14−6.55e13i)T2 1 + (1.10e5 - 6.24e5i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(−1.39e7+1.17e7i)T+(7.05e13−3.99e14i)T2 1 + (-1.39e7 + 1.17e7i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(1.69e7−9.76e6i)T+(3.22e14−5.59e14i)T2 1 + (1.69e7 - 9.76e6i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(1.76e7−3.06e7i)T+(−4.03e14−6.98e14i)T2 1 + (1.76e7 - 3.06e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(−2.71e7−2.28e7i)T+(2.63e14+1.49e15i)T2 1 + (-2.71e7 - 2.28e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−3.47e7+4.14e7i)T+(−3.91e14−2.21e15i)T2 1 + (-3.47e7 + 4.14e7i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(2.57e7+1.48e7i)T+(1.96e15+3.40e15i)T2 1 + (2.57e7 + 1.48e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(−8.11e7+2.95e7i)T+(6.00e15−5.03e15i)T2 1 + (-8.11e7 + 2.95e7i)T + (6.00e15 - 5.03e15i)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

−14.29651939360316398652390849140, −12.45043032521661001801405819008, −11.27874179408633172256377167115, −10.16391487307941816679624103374, −9.079421866196514128519270203668, −7.69774968055476450420074654349, −7.01586138692293955301541779012, −6.14261760170720886346441721552, −3.98791408827109681143831048686, −2.35239820711185662681905684326, 0.21007953167463042306836693702, 0.78311627405923296134633952433, 2.38000659553628796132504774013, 4.11149127065707569653667485349, 5.44140886049615132797335258069, 7.60533359107765838591026328623, 8.762873068577680250021165417937, 9.305534271879264782407175251619, 10.66291027753676219009780741043, 11.69129003811342745096947716693

Graph of the ZZ-function along the critical line