Properties

Label 2-3e5-27.22-c1-0-2
Degree $2$
Conductor $243$
Sign $0.0633 - 0.997i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.83 + 1.54i)2-s + (0.654 − 3.71i)4-s + (−0.0874 + 0.0318i)5-s + (−0.100 − 0.571i)7-s + (2.12 + 3.67i)8-s + (0.111 − 0.193i)10-s + (2.90 + 1.05i)11-s + (3.21 + 2.70i)13-s + (1.06 + 0.895i)14-s + (−2.50 − 0.909i)16-s + (0.995 − 1.72i)17-s + (1.92 + 3.33i)19-s + (0.0609 + 0.345i)20-s + (−6.98 + 2.54i)22-s + (−0.773 + 4.38i)23-s + ⋯
L(s)  = 1  + (−1.30 + 1.09i)2-s + (0.327 − 1.85i)4-s + (−0.0391 + 0.0142i)5-s + (−0.0380 − 0.215i)7-s + (0.750 + 1.29i)8-s + (0.0353 − 0.0612i)10-s + (0.876 + 0.318i)11-s + (0.892 + 0.749i)13-s + (0.285 + 0.239i)14-s + (−0.625 − 0.227i)16-s + (0.241 − 0.418i)17-s + (0.441 + 0.764i)19-s + (0.0136 + 0.0772i)20-s + (−1.48 + 0.541i)22-s + (−0.161 + 0.914i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496990 + 0.466437i\)
\(L(\frac12)\) \(\approx\) \(0.496990 + 0.466437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (1.83 - 1.54i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.0874 - 0.0318i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.100 + 0.571i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.90 - 1.05i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.21 - 2.70i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.995 + 1.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.773 - 4.38i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-4.90 + 4.11i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.287 + 1.63i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.01 - 3.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.839 + 0.704i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-6.48 - 2.36i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.623 - 3.53i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 5.40T + 53T^{2} \)
59 \( 1 + (-9.66 + 3.51i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.29 + 12.9i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.76 + 5.68i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.572 - 0.991i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0977 + 0.169i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.52 - 4.63i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (11.4 - 9.57i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-0.776 - 1.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.97 + 1.81i)T + (74.3 + 62.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.06152207778333097170907658205, −11.17336536219694513075278051839, −9.911980350733862223356409498742, −9.388410312394621374768499877872, −8.373399988210943018972754196308, −7.46287658891822861620444654190, −6.57655590905326580361685929964, −5.65530515379531852358401972732, −3.91690869741492047629805723770, −1.39107666701908968542744660339, 1.03320269830058611252362954666, 2.68966230909476029444390099888, 3.92314073856413809031403883548, 5.86220665383538287125846972012, 7.24391194082383949572370351833, 8.548248347758174998441820220975, 8.850829012412921192239902619899, 10.13587827257499205376175053688, 10.71933812074180430698193163316, 11.72933505703131663393027450176

Graph of the $Z$-function along the critical line