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Properties

Label 2-3e5-9.4-c1-0-1
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

\(L(1)\) \(\approx\) \(0.732816 - 0.266723i\)
\(L(\frac12)\) \(\approx\) \(0.732816 - 0.266723i\)
\(L(\frac{3}{2})\) 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

−11.67291277597440282345807792990, −11.27816545768228130079497670392, −9.970134498338217806747356838130, −9.442597974618904164443751839830, −8.359417877954847473222794628525, −7.58928588981198316701559322759, −5.69008708174375475843994123007, −4.13898603893715215567161453970, −2.81633874177436165822130979558, −1.58253159709117254658584109610, 0.989616248394752679227919534053, 4.07576078168250109027962060860, 5.30429964390919522704563660785, 6.48578776276259830121310862104, 7.25681678995039929360092217975, 8.317985277295488893953855231467, 8.829940353164016482865936054030, 10.13061490554711458803054309433, 10.82751211606044247092304697668, 12.17543091654032810082502758722

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