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Properties

Label 2-54-27.11-c8-0-18
Degree 22
Conductor 5454
Sign −0.987−0.156i-0.987 - 0.156i
Analytic cond. 21.998421.9984
Root an. cond. 4.690244.69024
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  + (7.27 − 8.66i)2-s + (−76.9 + 25.1i)3-s + (−22.2 − 126. i)4-s + (421. − 1.15e3i)5-s + (−341. + 850. i)6-s + (−165. + 937. i)7-s + (−1.25e3 − 724. i)8-s + (5.29e3 − 3.87e3i)9-s + (−6.96e3 − 1.20e4i)10-s + (−1.20e3 − 3.31e3i)11-s + (4.88e3 + 9.14e3i)12-s + (2.44e4 − 2.05e4i)13-s + (6.91e3 + 8.24e3i)14-s + (−3.27e3 + 9.96e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (−8.20e4 + 4.73e4i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.950 + 0.310i)3-s + (−0.0868 − 0.492i)4-s + (0.673 − 1.85i)5-s + (−0.263 + 0.656i)6-s + (−0.0688 + 0.390i)7-s + (−0.306 − 0.176i)8-s + (0.806 − 0.591i)9-s + (−0.696 − 1.20i)10-s + (−0.0825 − 0.226i)11-s + (0.235 + 0.440i)12-s + (0.855 − 0.717i)13-s + (0.180 + 0.214i)14-s + (−0.0647 + 1.96i)15-s + (−0.234 + 0.0855i)16-s + (−0.982 + 0.567i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.987−0.156i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((−0.987−0.156i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.987−0.156i-0.987 - 0.156i
Analytic conductor: 21.998421.9984
Root analytic conductor: 4.690244.69024
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ54(11,⋅)\chi_{54} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :4), −0.987−0.156i)(2,\ 54,\ (\ :4),\ -0.987 - 0.156i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.0972007+1.23735i0.0972007 + 1.23735i
L(12)L(\frac12) ≈\approx 0.0972007+1.23735i0.0972007 + 1.23735i
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)
bad2 1+(−7.27+8.66i)T 1 + (-7.27 + 8.66i)T
3 1+(76.9−25.1i)T 1 + (76.9 - 25.1i)T
good5 1+(−421.+1.15e3i)T+(−2.99e5−2.51e5i)T2 1 + (-421. + 1.15e3i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(165.−937.i)T+(−5.41e6−1.97e6i)T2 1 + (165. - 937. i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(1.20e3+3.31e3i)T+(−1.64e8+1.37e8i)T2 1 + (1.20e3 + 3.31e3i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(−2.44e4+2.05e4i)T+(1.41e8−8.03e8i)T2 1 + (-2.44e4 + 2.05e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(8.20e4−4.73e4i)T+(3.48e9−6.04e9i)T2 1 + (8.20e4 - 4.73e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(−4.97e4+8.62e4i)T+(−8.49e9−1.47e10i)T2 1 + (-4.97e4 + 8.62e4i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(4.73e5−8.34e4i)T+(7.35e10−2.67e10i)T2 1 + (4.73e5 - 8.34e4i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(2.47e5−2.95e5i)T+(−8.68e10−4.92e11i)T2 1 + (2.47e5 - 2.95e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(−1.36e5−7.72e5i)T+(−8.01e11+2.91e11i)T2 1 + (-1.36e5 - 7.72e5i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(9.49e5+1.64e6i)T+(−1.75e12+3.04e12i)T2 1 + (9.49e5 + 1.64e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(−9.60e5−1.14e6i)T+(−1.38e12+7.86e12i)T2 1 + (-9.60e5 - 1.14e6i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(−1.96e6+7.14e5i)T+(8.95e12−7.51e12i)T2 1 + (-1.96e6 + 7.14e5i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(1.52e6+2.68e5i)T+(2.23e13+8.14e12i)T2 1 + (1.52e6 + 2.68e5i)T + (2.23e13 + 8.14e12i)T^{2}
53 1+5.10e6iT−6.22e13T2 1 + 5.10e6iT - 6.22e13T^{2}
59 1+(6.95e6−1.91e7i)T+(−1.12e14−9.43e13i)T2 1 + (6.95e6 - 1.91e7i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(−4.65e6+2.63e7i)T+(−1.80e14−6.55e13i)T2 1 + (-4.65e6 + 2.63e7i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(−1.26e7+1.06e7i)T+(7.05e13−3.99e14i)T2 1 + (-1.26e7 + 1.06e7i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(1.46e7−8.44e6i)T+(3.22e14−5.59e14i)T2 1 + (1.46e7 - 8.44e6i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(1.24e7−2.15e7i)T+(−4.03e14−6.98e14i)T2 1 + (1.24e7 - 2.15e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(−2.32e7−1.95e7i)T+(2.63e14+1.49e15i)T2 1 + (-2.32e7 - 1.95e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−4.57e7+5.45e7i)T+(−3.91e14−2.21e15i)T2 1 + (-4.57e7 + 5.45e7i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(3.94e7+2.27e7i)T+(1.96e15+3.40e15i)T2 1 + (3.94e7 + 2.27e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(9.43e7−3.43e7i)T+(6.00e15−5.03e15i)T2 1 + (9.43e7 - 3.43e7i)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

−12.82289105966801995819595407283, −12.11455752121183591114139318506, −10.84120560534521377858541789621, −9.599014518042901858663153022191, −8.585829949516769315157033067192, −6.02621512984439236728738590781, −5.30527153557864507156758707914, −4.12673528423166762276872238221, −1.64960207423197928283419709087, −0.41269008338249187991376937497, 2.14443562550067043895376429121, 3.99864516019777268139554457741, 5.92078186184846482541973959957, 6.58317225023893518819144821950, 7.55090944932694686180012082600, 9.867759024436916672481215309665, 10.90139482001569140030664232022, 11.81598291597697802360038867237, 13.52012119870093442117439393203, 13.99154257163845500745944460511

Graph of the ZZ-function along the critical line