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Properties

Label 2-3e5-27.7-c1-0-3
Degree 22
Conductor 243243
Sign 0.769−0.638i0.769 - 0.638i
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  + (0.990 − 0.360i)2-s + (−0.680 + 0.571i)4-s + (0.303 + 1.71i)5-s + (1.88 + 1.58i)7-s + (−1.52 + 2.63i)8-s + (0.920 + 1.59i)10-s + (0.217 − 1.23i)11-s + (4.27 + 1.55i)13-s + (2.43 + 0.886i)14-s + (−0.249 + 1.41i)16-s + (−3.32 − 5.75i)17-s + (−0.124 + 0.215i)19-s + (−1.18 − 0.996i)20-s + (−0.229 − 1.30i)22-s + (−0.645 + 0.541i)23-s + ⋯
L(s)  = 1  + (0.700 − 0.254i)2-s + (−0.340 + 0.285i)4-s + (0.135 + 0.768i)5-s + (0.712 + 0.597i)7-s + (−0.538 + 0.932i)8-s + (0.291 + 0.504i)10-s + (0.0656 − 0.372i)11-s + (1.18 + 0.431i)13-s + (0.651 + 0.237i)14-s + (−0.0622 + 0.353i)16-s + (−0.806 − 1.39i)17-s + (−0.0285 + 0.0495i)19-s + (−0.265 − 0.222i)20-s + (−0.0489 − 0.277i)22-s + (−0.134 + 0.112i)23-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.769−0.638i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.769−0.638i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.769−0.638i0.769 - 0.638i
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), 0.769−0.638i)(2,\ 243,\ (\ :1/2),\ 0.769 - 0.638i)

Particular Values

L(1)L(1) ≈\approx 1.56445+0.564787i1.56445 + 0.564787i
L(12)L(\frac12) ≈\approx 1.56445+0.564787i1.56445 + 0.564787i
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+(−0.990+0.360i)T+(1.53−1.28i)T2 1 + (-0.990 + 0.360i)T + (1.53 - 1.28i)T^{2}
5 1+(−0.303−1.71i)T+(−4.69+1.71i)T2 1 + (-0.303 - 1.71i)T + (-4.69 + 1.71i)T^{2}
7 1+(−1.88−1.58i)T+(1.21+6.89i)T2 1 + (-1.88 - 1.58i)T + (1.21 + 6.89i)T^{2}
11 1+(−0.217+1.23i)T+(−10.3−3.76i)T2 1 + (-0.217 + 1.23i)T + (-10.3 - 3.76i)T^{2}
13 1+(−4.27−1.55i)T+(9.95+8.35i)T2 1 + (-4.27 - 1.55i)T + (9.95 + 8.35i)T^{2}
17 1+(3.32+5.75i)T+(−8.5+14.7i)T2 1 + (3.32 + 5.75i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.124−0.215i)T+(−9.5−16.4i)T2 1 + (0.124 - 0.215i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.645−0.541i)T+(3.99−22.6i)T2 1 + (0.645 - 0.541i)T + (3.99 - 22.6i)T^{2}
29 1+(0.481−0.175i)T+(22.2−18.6i)T2 1 + (0.481 - 0.175i)T + (22.2 - 18.6i)T^{2}
31 1+(0.628−0.527i)T+(5.38−30.5i)T2 1 + (0.628 - 0.527i)T + (5.38 - 30.5i)T^{2}
37 1+(1.30+2.25i)T+(−18.5+32.0i)T2 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2}
41 1+(7.66+2.78i)T+(31.4+26.3i)T2 1 + (7.66 + 2.78i)T + (31.4 + 26.3i)T^{2}
43 1+(−0.751+4.26i)T+(−40.4−14.7i)T2 1 + (-0.751 + 4.26i)T + (-40.4 - 14.7i)T^{2}
47 1+(−4.06−3.40i)T+(8.16+46.2i)T2 1 + (-4.06 - 3.40i)T + (8.16 + 46.2i)T^{2}
53 1−10.4T+53T2 1 - 10.4T + 53T^{2}
59 1+(−0.522−2.96i)T+(−55.4+20.1i)T2 1 + (-0.522 - 2.96i)T + (-55.4 + 20.1i)T^{2}
61 1+(−2.20−1.85i)T+(10.5+60.0i)T2 1 + (-2.20 - 1.85i)T + (10.5 + 60.0i)T^{2}
67 1+(9.47+3.44i)T+(51.3+43.0i)T2 1 + (9.47 + 3.44i)T + (51.3 + 43.0i)T^{2}
71 1+(−0.0447−0.0774i)T+(−35.5+61.4i)T2 1 + (-0.0447 - 0.0774i)T + (-35.5 + 61.4i)T^{2}
73 1+(−2.66+4.60i)T+(−36.5−63.2i)T2 1 + (-2.66 + 4.60i)T + (-36.5 - 63.2i)T^{2}
79 1+(4.48−1.63i)T+(60.5−50.7i)T2 1 + (4.48 - 1.63i)T + (60.5 - 50.7i)T^{2}
83 1+(−7.55+2.75i)T+(63.5−53.3i)T2 1 + (-7.55 + 2.75i)T + (63.5 - 53.3i)T^{2}
89 1+(−3.35+5.80i)T+(−44.5−77.0i)T2 1 + (-3.35 + 5.80i)T + (-44.5 - 77.0i)T^{2}
97 1+(−0.953+5.40i)T+(−91.1−33.1i)T2 1 + (-0.953 + 5.40i)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

−12.05058285331134209641046698509, −11.46978554060359509676891080826, −10.66140955286506617565192043765, −9.071191499000125101843658197238, −8.515565572753493246779592944754, −7.12508213595517298305687192546, −5.89433860834224475084403659748, −4.83744317397033787050424497051, −3.59044222523894312716472521921, −2.37834065101223502607858566078, 1.33458270119305769740771808623, 3.82333626792328727822694487069, 4.64361913737837403610789286196, 5.67315686015966979252044217455, 6.71276806145360121296662959480, 8.223190347454000023543721791592, 8.925097478717448609566783006698, 10.18565172299778347452590054204, 11.01265969196807131842359003907, 12.32164030032221860445737819113

Graph of the ZZ-function along the critical line