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Properties

Label 2-3e5-243.103-c1-0-9
Degree 22
Conductor 243243
Sign −0.318+0.947i-0.318 + 0.947i
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  + (−2.60 − 0.724i)2-s + (0.610 + 1.62i)3-s + (4.53 + 2.73i)4-s + (−1.88 + 0.0730i)5-s + (−0.415 − 4.65i)6-s + (−3.79 − 0.594i)7-s + (−6.10 − 6.46i)8-s + (−2.25 + 1.98i)9-s + (4.94 + 1.17i)10-s + (−0.256 − 1.87i)11-s + (−1.66 + 9.01i)12-s + (3.67 − 5.35i)13-s + (9.45 + 4.29i)14-s + (−1.26 − 3.00i)15-s + (6.25 + 11.8i)16-s + (3.18 − 1.59i)17-s + ⋯
L(s)  = 1  + (−1.83 − 0.512i)2-s + (0.352 + 0.935i)3-s + (2.26 + 1.36i)4-s + (−0.841 + 0.0326i)5-s + (−0.169 − 1.90i)6-s + (−1.43 − 0.224i)7-s + (−2.15 − 2.28i)8-s + (−0.751 + 0.660i)9-s + (1.56 + 0.370i)10-s + (−0.0774 − 0.566i)11-s + (−0.480 + 2.60i)12-s + (1.01 − 1.48i)13-s + (2.52 + 1.14i)14-s + (−0.327 − 0.775i)15-s + (1.56 + 2.96i)16-s + (0.771 − 0.387i)17-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((−0.318+0.947i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((−0.318+0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: −0.318+0.947i-0.318 + 0.947i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(103,⋅)\chi_{243} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), −0.318+0.947i)(2,\ 243,\ (\ :1/2),\ -0.318 + 0.947i)

Particular Values

L(1)L(1) ≈\approx 0.137142−0.190817i0.137142 - 0.190817i
L(12)L(\frac12) ≈\approx 0.137142−0.190817i0.137142 - 0.190817i
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+(−0.610−1.62i)T 1 + (-0.610 - 1.62i)T
good2 1+(2.60+0.724i)T+(1.71+1.03i)T2 1 + (2.60 + 0.724i)T + (1.71 + 1.03i)T^{2}
5 1+(1.88−0.0730i)T+(4.98−0.387i)T2 1 + (1.88 - 0.0730i)T + (4.98 - 0.387i)T^{2}
7 1+(3.79+0.594i)T+(6.66+2.13i)T2 1 + (3.79 + 0.594i)T + (6.66 + 2.13i)T^{2}
11 1+(0.256+1.87i)T+(−10.5+2.94i)T2 1 + (0.256 + 1.87i)T + (-10.5 + 2.94i)T^{2}
13 1+(−3.67+5.35i)T+(−4.68−12.1i)T2 1 + (-3.67 + 5.35i)T + (-4.68 - 12.1i)T^{2}
17 1+(−3.18+1.59i)T+(10.1−13.6i)T2 1 + (-3.18 + 1.59i)T + (10.1 - 13.6i)T^{2}
19 1+(0.0127+0.219i)T+(−18.8+2.20i)T2 1 + (0.0127 + 0.219i)T + (-18.8 + 2.20i)T^{2}
23 1+(−1.10+2.85i)T+(−17.0−15.4i)T2 1 + (-1.10 + 2.85i)T + (-17.0 - 15.4i)T^{2}
29 1+(−0.913−0.653i)T+(9.38+27.4i)T2 1 + (-0.913 - 0.653i)T + (9.38 + 27.4i)T^{2}
31 1+(−0.433+0.496i)T+(−4.19−30.7i)T2 1 + (-0.433 + 0.496i)T + (-4.19 - 30.7i)T^{2}
37 1+(3.46+4.65i)T+(−10.6+35.4i)T2 1 + (3.46 + 4.65i)T + (-10.6 + 35.4i)T^{2}
41 1+(0.698−2.71i)T+(−35.8−19.8i)T2 1 + (0.698 - 2.71i)T + (-35.8 - 19.8i)T^{2}
43 1+(8.61+7.81i)T+(4.16+42.7i)T2 1 + (8.61 + 7.81i)T + (4.16 + 42.7i)T^{2}
47 1+(3.40+3.90i)T+(−6.36+46.5i)T2 1 + (3.40 + 3.90i)T + (-6.36 + 46.5i)T^{2}
53 1+(1.02−5.78i)T+(−49.8−18.1i)T2 1 + (1.02 - 5.78i)T + (-49.8 - 18.1i)T^{2}
59 1+(6.66−2.72i)T+(42.1−41.3i)T2 1 + (6.66 - 2.72i)T + (42.1 - 41.3i)T^{2}
61 1+(−3.41+2.06i)T+(28.4−53.9i)T2 1 + (-3.41 + 2.06i)T + (28.4 - 53.9i)T^{2}
67 1+(6.03−4.31i)T+(21.6−63.3i)T2 1 + (6.03 - 4.31i)T + (21.6 - 63.3i)T^{2}
71 1+(3.34+11.1i)T+(−59.3+39.0i)T2 1 + (3.34 + 11.1i)T + (-59.3 + 39.0i)T^{2}
73 1+(11.2−2.66i)T+(65.2−32.7i)T2 1 + (11.2 - 2.66i)T + (65.2 - 32.7i)T^{2}
79 1+(0.372−0.365i)T+(1.53−78.9i)T2 1 + (0.372 - 0.365i)T + (1.53 - 78.9i)T^{2}
83 1+(−0.659−2.55i)T+(−72.6+40.0i)T2 1 + (-0.659 - 2.55i)T + (-72.6 + 40.0i)T^{2}
89 1+(0.664−2.21i)T+(−74.3−48.9i)T2 1 + (0.664 - 2.21i)T + (-74.3 - 48.9i)T^{2}
97 1+(−17.6−0.686i)T+(96.7+7.51i)T2 1 + (-17.6 - 0.686i)T + (96.7 + 7.51i)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

−11.33164354670492938667472889756, −10.44827331546384547070624654907, −10.04085785609870887212798848707, −8.937653077015617205653421855886, −8.285020265518967843708520402791, −7.36921801831389914826709349684, −5.98674154152495445395400267493, −3.48774101047068864139207620643, −3.10210692828041069478943617180, −0.32221983559311624037935492951, 1.59254559808787157173114408178, 3.29599366873943123928264400251, 6.13821745160568320622517278165, 6.72006962938510144768769019424, 7.58556064813340134431054813517, 8.477729039052764424583035789922, 9.262114951200628914694003028870, 10.05089169310510116685926319506, 11.45930879613384763667006499787, 11.99391193355370979821016736658

Graph of the ZZ-function along the critical line