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Properties

Label 2-3e5-9.4-c1-0-4
Degree $2$
Conductor $243$
Sign $-0.766 - 0.642i$
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  + (1.26 + 2.19i)2-s + (−2.20 + 3.82i)4-s + (0.233 − 0.405i)5-s + (1.61 + 2.79i)7-s − 6.10·8-s + 1.18·10-s + (−1.55 − 2.68i)11-s + (1.09 − 1.89i)13-s + (−4.08 + 7.07i)14-s + (−3.31 − 5.74i)16-s − 3·17-s + 0.0418·19-s + (1.03 + 1.78i)20-s + (3.93 − 6.81i)22-s + (3.05 − 5.28i)23-s + ⋯
L(s)  = 1  + (0.895 + 1.55i)2-s + (−1.10 + 1.91i)4-s + (0.104 − 0.181i)5-s + (0.609 + 1.05i)7-s − 2.15·8-s + 0.374·10-s + (−0.468 − 0.811i)11-s + (0.302 − 0.524i)13-s + (−1.09 + 1.89i)14-s + (−0.829 − 1.43i)16-s − 0.727·17-s + 0.00961·19-s + (0.230 + 0.399i)20-s + (0.838 − 1.45i)22-s + (0.636 − 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.638087 + 1.75313i\)
\(L(\frac12)\) \(\approx\) \(0.638087 + 1.75313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 - 2.19i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.233 + 0.405i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.61 - 2.79i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.55 + 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 + 1.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 0.0418T + 19T^{2} \)
23 \( 1 + (-3.05 + 5.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.28 - 5.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.11 + 5.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 + (-3.85 + 6.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.294 - 0.509i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.83 + 8.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 + (4.26 - 7.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.634 - 1.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.00 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 8.23T + 73T^{2} \)
79 \( 1 + (5.52 + 9.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.754 + 1.30i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + (9.32 + 16.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.92041198650417514818393849452, −11.86264274634770505873571901473, −10.72158128618991072194102765767, −8.816531664922877888942620926250, −8.535133747236343565010923630396, −7.37768713240127593633061704595, −6.18278929059991075745281497434, −5.43983327985074683071807670461, −4.56314641516219728333431657789, −2.94557159852271556906797993727, 1.43242677796416715617369024414, 2.83621914765407178998860006995, 4.28919918269374771642342202537, 4.81529407614241932614476437020, 6.43126229866415703057915634800, 7.79475196203073142913345498324, 9.346177572321196711634608607444, 10.23616214948378351657269331229, 10.95910834019060991258995965561, 11.62815522086861078283656525320

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