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Properties

Label 2-3e5-27.7-c1-0-5
Degree 22
Conductor 243243
Sign 0.0342+0.999i0.0342 + 0.999i
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  + (−0.390 + 0.142i)2-s + (−1.39 + 1.17i)4-s + (−0.384 − 2.18i)5-s + (−1.01 − 0.848i)7-s + (0.795 − 1.37i)8-s + (0.460 + 0.797i)10-s + (0.905 − 5.13i)11-s + (0.0169 + 0.00617i)13-s + (0.515 + 0.187i)14-s + (0.519 − 2.94i)16-s + (−1.56 − 2.71i)17-s + (−0.208 + 0.361i)19-s + (3.10 + 2.60i)20-s + (0.376 + 2.13i)22-s + (0.792 − 0.664i)23-s + ⋯
L(s)  = 1  + (−0.276 + 0.100i)2-s + (−0.699 + 0.587i)4-s + (−0.172 − 0.975i)5-s + (−0.382 − 0.320i)7-s + (0.281 − 0.486i)8-s + (0.145 + 0.252i)10-s + (0.273 − 1.54i)11-s + (0.00470 + 0.00171i)13-s + (0.137 + 0.0501i)14-s + (0.129 − 0.737i)16-s + (−0.379 − 0.658i)17-s + (−0.0478 + 0.0829i)19-s + (0.693 + 0.581i)20-s + (0.0802 + 0.454i)22-s + (0.165 − 0.138i)23-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.0342+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0342 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.0342+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0342 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.0342+0.999i0.0342 + 0.999i
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), 0.0342+0.999i)(2,\ 243,\ (\ :1/2),\ 0.0342 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.501784−0.484899i0.501784 - 0.484899i
L(12)L(\frac12) ≈\approx 0.501784−0.484899i0.501784 - 0.484899i
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+(0.390−0.142i)T+(1.53−1.28i)T2 1 + (0.390 - 0.142i)T + (1.53 - 1.28i)T^{2}
5 1+(0.384+2.18i)T+(−4.69+1.71i)T2 1 + (0.384 + 2.18i)T + (-4.69 + 1.71i)T^{2}
7 1+(1.01+0.848i)T+(1.21+6.89i)T2 1 + (1.01 + 0.848i)T + (1.21 + 6.89i)T^{2}
11 1+(−0.905+5.13i)T+(−10.3−3.76i)T2 1 + (-0.905 + 5.13i)T + (-10.3 - 3.76i)T^{2}
13 1+(−0.0169−0.00617i)T+(9.95+8.35i)T2 1 + (-0.0169 - 0.00617i)T + (9.95 + 8.35i)T^{2}
17 1+(1.56+2.71i)T+(−8.5+14.7i)T2 1 + (1.56 + 2.71i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.208−0.361i)T+(−9.5−16.4i)T2 1 + (0.208 - 0.361i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.792+0.664i)T+(3.99−22.6i)T2 1 + (-0.792 + 0.664i)T + (3.99 - 22.6i)T^{2}
29 1+(7.33−2.67i)T+(22.2−18.6i)T2 1 + (7.33 - 2.67i)T + (22.2 - 18.6i)T^{2}
31 1+(−2.85+2.39i)T+(5.38−30.5i)T2 1 + (-2.85 + 2.39i)T + (5.38 - 30.5i)T^{2}
37 1+(2.21+3.83i)T+(−18.5+32.0i)T2 1 + (2.21 + 3.83i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.45+1.25i)T+(31.4+26.3i)T2 1 + (3.45 + 1.25i)T + (31.4 + 26.3i)T^{2}
43 1+(1.44−8.18i)T+(−40.4−14.7i)T2 1 + (1.44 - 8.18i)T + (-40.4 - 14.7i)T^{2}
47 1+(−5.43−4.56i)T+(8.16+46.2i)T2 1 + (-5.43 - 4.56i)T + (8.16 + 46.2i)T^{2}
53 1−1.30T+53T2 1 - 1.30T + 53T^{2}
59 1+(−0.642−3.64i)T+(−55.4+20.1i)T2 1 + (-0.642 - 3.64i)T + (-55.4 + 20.1i)T^{2}
61 1+(−5.29−4.44i)T+(10.5+60.0i)T2 1 + (-5.29 - 4.44i)T + (10.5 + 60.0i)T^{2}
67 1+(−10.3−3.77i)T+(51.3+43.0i)T2 1 + (-10.3 - 3.77i)T + (51.3 + 43.0i)T^{2}
71 1+(3.04+5.26i)T+(−35.5+61.4i)T2 1 + (3.04 + 5.26i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.273+0.473i)T+(−36.5−63.2i)T2 1 + (-0.273 + 0.473i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.459−0.167i)T+(60.5−50.7i)T2 1 + (0.459 - 0.167i)T + (60.5 - 50.7i)T^{2}
83 1+(−4.33+1.57i)T+(63.5−53.3i)T2 1 + (-4.33 + 1.57i)T + (63.5 - 53.3i)T^{2}
89 1+(1.68−2.92i)T+(−44.5−77.0i)T2 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2}
97 1+(−1.72+9.79i)T+(−91.1−33.1i)T2 1 + (-1.72 + 9.79i)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

−11.96879328246082458064912780422, −10.97321967139074796256243485580, −9.618535768383830151142293331805, −8.856596666339093222517658407507, −8.227622197184078193691998977458, −7.04560758329660796389877974175, −5.59890686609059142331509056542, −4.39450301541084139166757451515, −3.32130927185436354277099830122, −0.63785935165237415554850599101, 2.06634732829371572326563103330, 3.81089266153517650433016824059, 5.06966530297209793477294156218, 6.38820306523184060508178536038, 7.30790505257607122214211524993, 8.626710211926974859544209617170, 9.630019641807091201905040793708, 10.27126383217846842353228214566, 11.18578277389389697904213226603, 12.34311400670985931596754848265

Graph of the ZZ-function along the critical line