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Properties

Label 2-3e4-27.11-c8-0-12
Degree 22
Conductor 8181
Sign 0.654−0.756i0.654 - 0.756i
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  + (−4.74 + 5.66i)2-s + (34.9 + 198. i)4-s + (−15.1 + 41.5i)5-s + (243. − 1.38e3i)7-s + (−2.92e3 − 1.68e3i)8-s + (−163. − 283. i)10-s + (4.42e3 + 1.21e4i)11-s + (4.32e4 − 3.62e4i)13-s + (6.66e3 + 7.94e3i)14-s + (−2.49e4 + 9.09e3i)16-s + (2.65e4 − 1.53e4i)17-s + (6.82e4 − 1.18e5i)19-s + (−8.77e3 − 1.54e3i)20-s + (−8.97e4 − 3.26e4i)22-s + (−4.12e5 + 7.27e4i)23-s + ⋯
L(s)  = 1  + (−0.296 + 0.353i)2-s + (0.136 + 0.774i)4-s + (−0.0242 + 0.0664i)5-s + (0.101 − 0.575i)7-s + (−0.714 − 0.412i)8-s + (−0.0163 − 0.0283i)10-s + (0.302 + 0.829i)11-s + (1.51 − 1.26i)13-s + (0.173 + 0.206i)14-s + (−0.381 + 0.138i)16-s + (0.317 − 0.183i)17-s + (0.523 − 0.906i)19-s + (−0.0548 − 0.00966i)20-s + (−0.383 − 0.139i)22-s + (−1.47 + 0.260i)23-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.654−0.756i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+4) L(s)=((0.654−0.756i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.654−0.756i0.654 - 0.756i
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), 0.654−0.756i)(2,\ 81,\ (\ :4),\ 0.654 - 0.756i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.68191+0.768992i1.68191 + 0.768992i
L(12)L(\frac12) ≈\approx 1.68191+0.768992i1.68191 + 0.768992i
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+(4.74−5.66i)T+(−44.4−252.i)T2 1 + (4.74 - 5.66i)T + (-44.4 - 252. i)T^{2}
5 1+(15.1−41.5i)T+(−2.99e5−2.51e5i)T2 1 + (15.1 - 41.5i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(−243.+1.38e3i)T+(−5.41e6−1.97e6i)T2 1 + (-243. + 1.38e3i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(−4.42e3−1.21e4i)T+(−1.64e8+1.37e8i)T2 1 + (-4.42e3 - 1.21e4i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(−4.32e4+3.62e4i)T+(1.41e8−8.03e8i)T2 1 + (-4.32e4 + 3.62e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(−2.65e4+1.53e4i)T+(3.48e9−6.04e9i)T2 1 + (-2.65e4 + 1.53e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(−6.82e4+1.18e5i)T+(−8.49e9−1.47e10i)T2 1 + (-6.82e4 + 1.18e5i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(4.12e5−7.27e4i)T+(7.35e10−2.67e10i)T2 1 + (4.12e5 - 7.27e4i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(−2.50e5+2.98e5i)T+(−8.68e10−4.92e11i)T2 1 + (-2.50e5 + 2.98e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(2.55e4+1.44e5i)T+(−8.01e11+2.91e11i)T2 1 + (2.55e4 + 1.44e5i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(−8.98e5−1.55e6i)T+(−1.75e12+3.04e12i)T2 1 + (-8.98e5 - 1.55e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(−2.27e6−2.70e6i)T+(−1.38e12+7.86e12i)T2 1 + (-2.27e6 - 2.70e6i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(−3.81e5+1.38e5i)T+(8.95e12−7.51e12i)T2 1 + (-3.81e5 + 1.38e5i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(−4.80e6−8.47e5i)T+(2.23e13+8.14e12i)T2 1 + (-4.80e6 - 8.47e5i)T + (2.23e13 + 8.14e12i)T^{2}
53 1−1.24e7iT−6.22e13T2 1 - 1.24e7iT - 6.22e13T^{2}
59 1+(3.55e6−9.77e6i)T+(−1.12e14−9.43e13i)T2 1 + (3.55e6 - 9.77e6i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(−4.66e5+2.64e6i)T+(−1.80e14−6.55e13i)T2 1 + (-4.66e5 + 2.64e6i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(−1.30e7+1.09e7i)T+(7.05e13−3.99e14i)T2 1 + (-1.30e7 + 1.09e7i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(−6.93e6+4.00e6i)T+(3.22e14−5.59e14i)T2 1 + (-6.93e6 + 4.00e6i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(1.28e7−2.22e7i)T+(−4.03e14−6.98e14i)T2 1 + (1.28e7 - 2.22e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(−1.24e6−1.04e6i)T+(2.63e14+1.49e15i)T2 1 + (-1.24e6 - 1.04e6i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−4.09e7+4.87e7i)T+(−3.91e14−2.21e15i)T2 1 + (-4.09e7 + 4.87e7i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(2.75e7+1.59e7i)T+(1.96e15+3.40e15i)T2 1 + (2.75e7 + 1.59e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(−4.45e7+1.62e7i)T+(6.00e15−5.03e15i)T2 1 + (-4.45e7 + 1.62e7i)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

−12.90035096427852567829877992594, −11.79892678661785927089600415677, −10.65797080613425815959551271174, −9.358786640017528761882979422732, −8.118225264542351991329123160967, −7.30945099294214053784072573353, −6.03141267231588112186236877236, −4.19859669245610767229059253953, −2.97372675544593347402572166383, −0.936187772644156042239889112384, 0.886435572747332940998375591979, 2.06346282816438901950196105105, 3.82851151509504262638102106832, 5.64869842785008657957317134007, 6.42126710199653072623658982535, 8.398801339854974229299883685554, 9.177052624571255574103790397000, 10.44256757204319236397475897134, 11.38370800435565837793690010663, 12.23535044362979697893747994914

Graph of the ZZ-function along the critical line