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Properties

Label 2-18-9.4-c5-0-0
Degree 22
Conductor 1818
Sign −0.754−0.656i-0.754 - 0.656i
Analytic cond. 2.886902.88690
Root an. cond. 1.699091.69909
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−14.6 + 5.19i)3-s + (−7.99 + 13.8i)4-s + (−28.1 + 48.8i)5-s + (−47.3 − 40.5i)6-s + (11.1 + 19.3i)7-s − 63.9·8-s + (189 − 152. i)9-s − 225.·10-s + (296. + 513. i)11-s + (45.5 − 245. i)12-s + (129. − 224. i)13-s + (−44.6 + 77.2i)14-s + (160. − 864. i)15-s + (−128 − 221. i)16-s + 1.62e3·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.942 + 0.333i)3-s + (−0.249 + 0.433i)4-s + (−0.504 + 0.873i)5-s + (−0.537 − 0.459i)6-s + (0.0860 + 0.148i)7-s − 0.353·8-s + (0.777 − 0.628i)9-s − 0.713·10-s + (0.738 + 1.27i)11-s + (0.0913 − 0.491i)12-s + (0.212 − 0.368i)13-s + (−0.0608 + 0.105i)14-s + (0.184 − 0.991i)15-s + (−0.125 − 0.216i)16-s + 1.36·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.754−0.656i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+5/2) L(s)=((−0.754−0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.754−0.656i-0.754 - 0.656i
Analytic conductor: 2.886902.88690
Root analytic conductor: 1.699091.69909
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ18(13,⋅)\chi_{18} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :5/2), −0.754−0.656i)(2,\ 18,\ (\ :5/2),\ -0.754 - 0.656i)

Particular Values

L(3)L(3) ≈\approx 0.358642+0.957765i0.358642 + 0.957765i
L(12)L(\frac12) ≈\approx 0.358642+0.957765i0.358642 + 0.957765i
L(72)L(\frac{7}{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)
bad2 1+(−2−3.46i)T 1 + (-2 - 3.46i)T
3 1+(14.6−5.19i)T 1 + (14.6 - 5.19i)T
good5 1+(28.1−48.8i)T+(−1.56e3−2.70e3i)T2 1 + (28.1 - 48.8i)T + (-1.56e3 - 2.70e3i)T^{2}
7 1+(−11.1−19.3i)T+(−8.40e3+1.45e4i)T2 1 + (-11.1 - 19.3i)T + (-8.40e3 + 1.45e4i)T^{2}
11 1+(−296.−513.i)T+(−8.05e4+1.39e5i)T2 1 + (-296. - 513. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+(−129.+224.i)T+(−1.85e5−3.21e5i)T2 1 + (-129. + 224. i)T + (-1.85e5 - 3.21e5i)T^{2}
17 1−1.62e3T+1.41e6T2 1 - 1.62e3T + 1.41e6T^{2}
19 1+2.64e3T+2.47e6T2 1 + 2.64e3T + 2.47e6T^{2}
23 1+(1.56e3−2.71e3i)T+(−3.21e6−5.57e6i)T2 1 + (1.56e3 - 2.71e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+(62.2+107.i)T+(−1.02e7+1.77e7i)T2 1 + (62.2 + 107. i)T + (-1.02e7 + 1.77e7i)T^{2}
31 1+(−4.37e3+7.57e3i)T+(−1.43e7−2.47e7i)T2 1 + (-4.37e3 + 7.57e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1−9.74e3T+6.93e7T2 1 - 9.74e3T + 6.93e7T^{2}
41 1+(−3.73e3+6.47e3i)T+(−5.79e7−1.00e8i)T2 1 + (-3.73e3 + 6.47e3i)T + (-5.79e7 - 1.00e8i)T^{2}
43 1+(−6.25e3−1.08e4i)T+(−7.35e7+1.27e8i)T2 1 + (-6.25e3 - 1.08e4i)T + (-7.35e7 + 1.27e8i)T^{2}
47 1+(−4.40e3−7.63e3i)T+(−1.14e8+1.98e8i)T2 1 + (-4.40e3 - 7.63e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+1.07e4T+4.18e8T2 1 + 1.07e4T + 4.18e8T^{2}
59 1+(−7.77e3+1.34e4i)T+(−3.57e8−6.19e8i)T2 1 + (-7.77e3 + 1.34e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−5.96e3−1.03e4i)T+(−4.22e8+7.31e8i)T2 1 + (-5.96e3 - 1.03e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(3.08e3−5.35e3i)T+(−6.75e8−1.16e9i)T2 1 + (3.08e3 - 5.35e3i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+1.04e4T+1.80e9T2 1 + 1.04e4T + 1.80e9T^{2}
73 1+3.83e4T+2.07e9T2 1 + 3.83e4T + 2.07e9T^{2}
79 1+(8.92e3+1.54e4i)T+(−1.53e9+2.66e9i)T2 1 + (8.92e3 + 1.54e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+(−2.16e4−3.74e4i)T+(−1.96e9+3.41e9i)T2 1 + (-2.16e4 - 3.74e4i)T + (-1.96e9 + 3.41e9i)T^{2}
89 1+9.40e4T+5.58e9T2 1 + 9.40e4T + 5.58e9T^{2}
97 1+(−4.07e4−7.05e4i)T+(−4.29e9+7.43e9i)T2 1 + (-4.07e4 - 7.05e4i)T + (-4.29e9 + 7.43e9i)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

−17.80723027863276360141257158981, −16.85083147702780095971127397093, −15.37356826798086773340906034448, −14.71215661462253925266403506559, −12.62411716956373630162670812521, −11.45150232959813451227415718304, −9.893578548057804709635961400549, −7.48744271895944263411487098725, −6.08349614267157565472955749215, −4.13378035640076703048240740136, 0.874351368839317877182743688301, 4.34834546863170708381835743615, 6.13022856964328002006875292662, 8.486321866727081155140317189704, 10.57559409097599858330115099374, 11.83865182996358570874042275611, 12.69221572029100091838100060267, 14.18963639247044396136160532301, 16.21422977660534182139442444076, 16.98047680108832992097513354983

Graph of the ZZ-function along the critical line