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Properties

Label 2-3e5-27.22-c1-0-5
Degree 22
Conductor 243243
Sign 0.441+0.897i0.441 + 0.897i
Analytic cond. 1.940361.94036
Root an. cond. 1.392961.39296
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 1.08i)2-s + (0.143 − 0.812i)4-s + (−1.06 + 0.386i)5-s + (−0.678 − 3.84i)7-s + (−0.987 − 1.70i)8-s + (0.949 − 1.64i)10-s + (−1.75 − 0.639i)11-s + (−0.561 − 0.470i)13-s + (5.03 + 4.22i)14-s + (4.66 + 1.69i)16-s + (0.944 − 1.63i)17-s + (−1.37 − 2.37i)19-s + (0.161 + 0.918i)20-s + (2.95 − 1.07i)22-s + (1.01 − 5.73i)23-s + ⋯
L(s)  = 1  + (−0.910 + 0.764i)2-s + (0.0716 − 0.406i)4-s + (−0.474 + 0.172i)5-s + (−0.256 − 1.45i)7-s + (−0.349 − 0.604i)8-s + (0.300 − 0.519i)10-s + (−0.529 − 0.192i)11-s + (−0.155 − 0.130i)13-s + (1.34 + 1.12i)14-s + (1.16 + 0.424i)16-s + (0.229 − 0.396i)17-s + (−0.314 − 0.544i)19-s + (0.0362 + 0.205i)20-s + (0.629 − 0.229i)22-s + (0.211 − 1.19i)23-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.441+0.897i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.441+0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.441+0.897i0.441 + 0.897i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(109,⋅)\chi_{243} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 0.441+0.897i)(2,\ 243,\ (\ :1/2),\ 0.441 + 0.897i)

Particular Values

L(1)L(1) ≈\approx 0.355161−0.220977i0.355161 - 0.220977i
L(12)L(\frac12) ≈\approx 0.355161−0.220977i0.355161 - 0.220977i
L(32)L(\frac{3}{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)
bad3 1 1
good2 1+(1.28−1.08i)T+(0.347−1.96i)T2 1 + (1.28 - 1.08i)T + (0.347 - 1.96i)T^{2}
5 1+(1.06−0.386i)T+(3.83−3.21i)T2 1 + (1.06 - 0.386i)T + (3.83 - 3.21i)T^{2}
7 1+(0.678+3.84i)T+(−6.57+2.39i)T2 1 + (0.678 + 3.84i)T + (-6.57 + 2.39i)T^{2}
11 1+(1.75+0.639i)T+(8.42+7.07i)T2 1 + (1.75 + 0.639i)T + (8.42 + 7.07i)T^{2}
13 1+(0.561+0.470i)T+(2.25+12.8i)T2 1 + (0.561 + 0.470i)T + (2.25 + 12.8i)T^{2}
17 1+(−0.944+1.63i)T+(−8.5−14.7i)T2 1 + (-0.944 + 1.63i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.37+2.37i)T+(−9.5+16.4i)T2 1 + (1.37 + 2.37i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.01+5.73i)T+(−21.6−7.86i)T2 1 + (-1.01 + 5.73i)T + (-21.6 - 7.86i)T^{2}
29 1+(−4.07+3.41i)T+(5.03−28.5i)T2 1 + (-4.07 + 3.41i)T + (5.03 - 28.5i)T^{2}
31 1+(−0.232+1.32i)T+(−29.1−10.6i)T2 1 + (-0.232 + 1.32i)T + (-29.1 - 10.6i)T^{2}
37 1+(1.69−2.94i)T+(−18.5−32.0i)T2 1 + (1.69 - 2.94i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.37+1.15i)T+(7.11+40.3i)T2 1 + (1.37 + 1.15i)T + (7.11 + 40.3i)T^{2}
43 1+(4.72+1.72i)T+(32.9+27.6i)T2 1 + (4.72 + 1.72i)T + (32.9 + 27.6i)T^{2}
47 1+(0.296+1.68i)T+(−44.1+16.0i)T2 1 + (0.296 + 1.68i)T + (-44.1 + 16.0i)T^{2}
53 1+2.84T+53T2 1 + 2.84T + 53T^{2}
59 1+(10.5−3.85i)T+(45.1−37.9i)T2 1 + (10.5 - 3.85i)T + (45.1 - 37.9i)T^{2}
61 1+(−0.908−5.15i)T+(−57.3+20.8i)T2 1 + (-0.908 - 5.15i)T + (-57.3 + 20.8i)T^{2}
67 1+(1.44+1.21i)T+(11.6+65.9i)T2 1 + (1.44 + 1.21i)T + (11.6 + 65.9i)T^{2}
71 1+(6.09−10.5i)T+(−35.5−61.4i)T2 1 + (6.09 - 10.5i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.94+8.56i)T+(−36.5+63.2i)T2 1 + (4.94 + 8.56i)T + (-36.5 + 63.2i)T^{2}
79 1+(−9.46+7.94i)T+(13.7−77.7i)T2 1 + (-9.46 + 7.94i)T + (13.7 - 77.7i)T^{2}
83 1+(−8.94+7.50i)T+(14.4−81.7i)T2 1 + (-8.94 + 7.50i)T + (14.4 - 81.7i)T^{2}
89 1+(2.86+4.96i)T+(−44.5+77.0i)T2 1 + (2.86 + 4.96i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.322−0.117i)T+(74.3+62.3i)T2 1 + (-0.322 - 0.117i)T + (74.3 + 62.3i)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

−11.83527445363922214162976005963, −10.59169250906027462247242458472, −10.03027680135126260814130667963, −8.821865856710106179879164587875, −7.82273926589305693046517265365, −7.22503248439485818392171600851, −6.31236713689857098066005197583, −4.54047622028121367613182841715, −3.27146304982485957276178069209, −0.44297277286609786219492046567, 1.90744987158506716307283312505, 3.19344053498557107211273077074, 5.08854050437246537231941397002, 6.10917941452797856279594592822, 7.81705263724768649924559976293, 8.591708461327995859450803853875, 9.425798168419728790141277216217, 10.26216314833348603813896736810, 11.31642430904784410214084641921, 12.11204465041396024283580822223

Graph of the ZZ-function along the critical line