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Properties

Label 2-3e4-27.11-c8-0-22
Degree 22
Conductor 8181
Sign −0.707−0.706i-0.707 - 0.706i
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  + (19.4 − 23.2i)2-s + (−115. − 652. i)4-s + (197. − 542. i)5-s + (379. − 2.15e3i)7-s + (−1.06e4 − 6.15e3i)8-s + (−8.74e3 − 1.51e4i)10-s + (3.14e3 + 8.63e3i)11-s + (−2.15e4 + 1.80e4i)13-s + (−4.25e4 − 5.07e4i)14-s + (−1.91e5 + 6.96e4i)16-s + (1.18e5 − 6.86e4i)17-s + (9.47e4 − 1.64e5i)19-s + (−3.76e5 − 6.63e4i)20-s + (2.61e5 + 9.52e4i)22-s + (−2.06e5 + 3.64e4i)23-s + ⋯
L(s)  = 1  + (1.21 − 1.45i)2-s + (−0.449 − 2.54i)4-s + (0.315 − 0.867i)5-s + (0.158 − 0.896i)7-s + (−2.60 − 1.50i)8-s + (−0.874 − 1.51i)10-s + (0.214 + 0.589i)11-s + (−0.754 + 0.633i)13-s + (−1.10 − 1.32i)14-s + (−2.91 + 1.06i)16-s + (1.42 − 0.822i)17-s + (0.727 − 1.25i)19-s + (−2.35 − 0.414i)20-s + (1.11 + 0.406i)22-s + (−0.739 + 0.130i)23-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((−0.707−0.706i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+4) L(s)=((−0.707−0.706i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: −0.707−0.706i-0.707 - 0.706i
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), −0.707−0.706i)(2,\ 81,\ (\ :4),\ -0.707 - 0.706i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.36867+3.30985i1.36867 + 3.30985i
L(12)L(\frac12) ≈\approx 1.36867+3.30985i1.36867 + 3.30985i
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+(−19.4+23.2i)T+(−44.4−252.i)T2 1 + (-19.4 + 23.2i)T + (-44.4 - 252. i)T^{2}
5 1+(−197.+542.i)T+(−2.99e5−2.51e5i)T2 1 + (-197. + 542. i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(−379.+2.15e3i)T+(−5.41e6−1.97e6i)T2 1 + (-379. + 2.15e3i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(−3.14e3−8.63e3i)T+(−1.64e8+1.37e8i)T2 1 + (-3.14e3 - 8.63e3i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(2.15e4−1.80e4i)T+(1.41e8−8.03e8i)T2 1 + (2.15e4 - 1.80e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(−1.18e5+6.86e4i)T+(3.48e9−6.04e9i)T2 1 + (-1.18e5 + 6.86e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(−9.47e4+1.64e5i)T+(−8.49e9−1.47e10i)T2 1 + (-9.47e4 + 1.64e5i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(2.06e5−3.64e4i)T+(7.35e10−2.67e10i)T2 1 + (2.06e5 - 3.64e4i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(5.59e5−6.66e5i)T+(−8.68e10−4.92e11i)T2 1 + (5.59e5 - 6.66e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(−2.01e5−1.14e6i)T+(−8.01e11+2.91e11i)T2 1 + (-2.01e5 - 1.14e6i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(1.07e5+1.86e5i)T+(−1.75e12+3.04e12i)T2 1 + (1.07e5 + 1.86e5i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(−8.39e5−1.00e6i)T+(−1.38e12+7.86e12i)T2 1 + (-8.39e5 - 1.00e6i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(−1.24e6+4.52e5i)T+(8.95e12−7.51e12i)T2 1 + (-1.24e6 + 4.52e5i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(3.65e6+6.45e5i)T+(2.23e13+8.14e12i)T2 1 + (3.65e6 + 6.45e5i)T + (2.23e13 + 8.14e12i)T^{2}
53 1+2.09e6iT−6.22e13T2 1 + 2.09e6iT - 6.22e13T^{2}
59 1+(−4.17e6+1.14e7i)T+(−1.12e14−9.43e13i)T2 1 + (-4.17e6 + 1.14e7i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(−4.25e6+2.41e7i)T+(−1.80e14−6.55e13i)T2 1 + (-4.25e6 + 2.41e7i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(−8.20e6+6.88e6i)T+(7.05e13−3.99e14i)T2 1 + (-8.20e6 + 6.88e6i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(8.38e6−4.84e6i)T+(3.22e14−5.59e14i)T2 1 + (8.38e6 - 4.84e6i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(−9.79e5+1.69e6i)T+(−4.03e14−6.98e14i)T2 1 + (-9.79e5 + 1.69e6i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(1.87e7+1.56e7i)T+(2.63e14+1.49e15i)T2 1 + (1.87e7 + 1.56e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−6.79e5+8.09e5i)T+(−3.91e14−2.21e15i)T2 1 + (-6.79e5 + 8.09e5i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(1.03e8+5.96e7i)T+(1.96e15+3.40e15i)T2 1 + (1.03e8 + 5.96e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(−1.40e8+5.11e7i)T+(6.00e15−5.03e15i)T2 1 + (-1.40e8 + 5.11e7i)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

−12.17275821965303080735306955076, −11.25388958630255604249312156407, −9.983017231520414986559904167166, −9.329073502385749063674351347500, −7.10526077470444585739525404674, −5.25164798947972003864593813825, −4.64433563853971428724015947957, −3.28008913234175000672333321688, −1.72310837247735969649791846583, −0.73169136046461978916422434556, 2.68865539220535841456527731113, 3.85165423392964409389328780960, 5.68749804004074500174626863805, 5.91816555274751111580991976571, 7.48307641823128658272357930944, 8.266098779992328556506266818508, 9.954124651686346393827845320288, 11.78001515880860787772136655590, 12.53710749084476885565882039562, 13.77525107481611601632030765714

Graph of the ZZ-function along the critical line