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Properties

Label 2-3e5-27.7-c1-0-4
Degree 22
Conductor 243243
Sign 0.690+0.723i0.690 + 0.723i
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  + (−2.25 + 0.821i)2-s + (2.88 − 2.42i)4-s + (−0.0161 − 0.0916i)5-s + (−0.444 − 0.372i)7-s + (−2.12 + 3.67i)8-s + (0.111 + 0.193i)10-s + (0.537 − 3.04i)11-s + (−3.94 − 1.43i)13-s + (1.30 + 0.476i)14-s + (0.462 − 2.62i)16-s + (−0.995 − 1.72i)17-s + (1.92 − 3.33i)19-s + (−0.268 − 0.225i)20-s + (1.28 + 7.31i)22-s + (3.41 − 2.86i)23-s + ⋯
L(s)  = 1  + (−1.59 + 0.580i)2-s + (1.44 − 1.21i)4-s + (−0.00722 − 0.0409i)5-s + (−0.167 − 0.140i)7-s + (−0.750 + 1.29i)8-s + (0.0353 + 0.0612i)10-s + (0.161 − 0.918i)11-s + (−1.09 − 0.398i)13-s + (0.349 + 0.127i)14-s + (0.115 − 0.655i)16-s + (−0.241 − 0.418i)17-s + (0.441 − 0.764i)19-s + (−0.0600 − 0.0504i)20-s + (0.275 + 1.55i)22-s + (0.711 − 0.596i)23-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.690+0.723i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.690+0.723i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.690+0.723i0.690 + 0.723i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(55,⋅)\chi_{243} (55, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 0.690+0.723i)(2,\ 243,\ (\ :1/2),\ 0.690 + 0.723i)

Particular Values

L(1)L(1) ≈\approx 0.443536−0.189949i0.443536 - 0.189949i
L(12)L(\frac12) ≈\approx 0.443536−0.189949i0.443536 - 0.189949i
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+(2.25−0.821i)T+(1.53−1.28i)T2 1 + (2.25 - 0.821i)T + (1.53 - 1.28i)T^{2}
5 1+(0.0161+0.0916i)T+(−4.69+1.71i)T2 1 + (0.0161 + 0.0916i)T + (-4.69 + 1.71i)T^{2}
7 1+(0.444+0.372i)T+(1.21+6.89i)T2 1 + (0.444 + 0.372i)T + (1.21 + 6.89i)T^{2}
11 1+(−0.537+3.04i)T+(−10.3−3.76i)T2 1 + (-0.537 + 3.04i)T + (-10.3 - 3.76i)T^{2}
13 1+(3.94+1.43i)T+(9.95+8.35i)T2 1 + (3.94 + 1.43i)T + (9.95 + 8.35i)T^{2}
17 1+(0.995+1.72i)T+(−8.5+14.7i)T2 1 + (0.995 + 1.72i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.92+3.33i)T+(−9.5−16.4i)T2 1 + (-1.92 + 3.33i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.41+2.86i)T+(3.99−22.6i)T2 1 + (-3.41 + 2.86i)T + (3.99 - 22.6i)T^{2}
29 1+(−6.01+2.18i)T+(22.2−18.6i)T2 1 + (-6.01 + 2.18i)T + (22.2 - 18.6i)T^{2}
31 1+(−1.26+1.06i)T+(5.38−30.5i)T2 1 + (-1.26 + 1.06i)T + (5.38 - 30.5i)T^{2}
37 1+(2.01+3.49i)T+(−18.5+32.0i)T2 1 + (2.01 + 3.49i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.03+0.374i)T+(31.4+26.3i)T2 1 + (1.03 + 0.374i)T + (31.4 + 26.3i)T^{2}
43 1+(1.19−6.79i)T+(−40.4−14.7i)T2 1 + (1.19 - 6.79i)T + (-40.4 - 14.7i)T^{2}
47 1+(2.75+2.30i)T+(8.16+46.2i)T2 1 + (2.75 + 2.30i)T + (8.16 + 46.2i)T^{2}
53 1+5.40T+53T2 1 + 5.40T + 53T^{2}
59 1+(−1.78−10.1i)T+(−55.4+20.1i)T2 1 + (-1.78 - 10.1i)T + (-55.4 + 20.1i)T^{2}
61 1+(10.1+8.48i)T+(10.5+60.0i)T2 1 + (10.1 + 8.48i)T + (10.5 + 60.0i)T^{2}
67 1+(−8.30−3.02i)T+(51.3+43.0i)T2 1 + (-8.30 - 3.02i)T + (51.3 + 43.0i)T^{2}
71 1+(−0.572−0.991i)T+(−35.5+61.4i)T2 1 + (-0.572 - 0.991i)T + (-35.5 + 61.4i)T^{2}
73 1+(0.0977−0.169i)T+(−36.5−63.2i)T2 1 + (0.0977 - 0.169i)T + (-36.5 - 63.2i)T^{2}
79 1+(−6.77+2.46i)T+(60.5−50.7i)T2 1 + (-6.77 + 2.46i)T + (60.5 - 50.7i)T^{2}
83 1+(14.0−5.09i)T+(63.5−53.3i)T2 1 + (14.0 - 5.09i)T + (63.5 - 53.3i)T^{2}
89 1+(0.776−1.34i)T+(−44.5−77.0i)T2 1 + (0.776 - 1.34i)T + (-44.5 - 77.0i)T^{2}
97 1+(−0.919+5.21i)T+(−91.1−33.1i)T2 1 + (-0.919 + 5.21i)T + (-91.1 - 33.1i)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

−11.59238382681422858672527114031, −10.70631629793351394543494549923, −9.860248757774946959917547785024, −8.989681946243214373687135142423, −8.196435808748084010364826132712, −7.15963846597781706516698739374, −6.38689059319190092766324190923, −4.94339336540978008625501247509, −2.77297374496445432863010632444, −0.66830618715030517674368594508, 1.62850608450043783256394884498, 3.01111536636660680953856431517, 4.87111144662319341533896607654, 6.72938766435507366126183420782, 7.49179258777373113565046847428, 8.568445248080881548065742539771, 9.492762677903980064171253932213, 10.08578444066471908712075763994, 11.02655878978542289738971945271, 12.06828444974761772568095486327

Graph of the ZZ-function along the critical line