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Properties

Label 2-3e5-27.7-c1-0-4
Degree $2$
Conductor $243$
Sign $0.690 + 0.723i$
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  + (−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

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

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443536 - 0.189949i\)
\(L(\frac12)\) \(\approx\) \(0.443536 - 0.189949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 - 0.821i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.0161 + 0.0916i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.444 + 0.372i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.537 + 3.04i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (3.94 + 1.43i)T + (9.95 + 8.35i)T^{2} \)
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 \( 1 + (-3.41 + 2.86i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.01 + 2.18i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.26 + 1.06i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.01 + 3.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.03 + 0.374i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.19 - 6.79i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.75 + 2.30i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 5.40T + 53T^{2} \)
59 \( 1 + (-1.78 - 10.1i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.1 + 8.48i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.30 - 3.02i)T + (51.3 + 43.0i)T^{2} \)
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 \( 1 + (-6.77 + 2.46i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (14.0 - 5.09i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (0.776 - 1.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.919 + 5.21i)T + (-91.1 - 33.1i)T^{2} \)
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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

−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 $Z$-function along the critical line