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Properties

Label 2-3e5-9.4-c1-0-3
Degree $2$
Conductor $243$
Sign $0.173 - 0.984i$
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  + (0.439 + 0.761i)2-s + (0.613 − 1.06i)4-s + (−1.93 + 3.35i)5-s + (1.09 + 1.89i)7-s + 2.83·8-s − 3.41·10-s + (−0.0812 − 0.140i)11-s + (−1.20 + 2.08i)13-s + (−0.960 + 1.66i)14-s + (0.0209 + 0.0362i)16-s + 3·17-s + 3.59·19-s + (2.37 + 4.12i)20-s + (0.0714 − 0.123i)22-s + (−1.41 + 2.45i)23-s + ⋯
L(s)  = 1  + (0.310 + 0.538i)2-s + (0.306 − 0.531i)4-s + (−0.867 + 1.50i)5-s + (0.412 + 0.715i)7-s + 1.00·8-s − 1.07·10-s + (−0.0244 − 0.0424i)11-s + (−0.334 + 0.579i)13-s + (−0.256 + 0.444i)14-s + (0.00523 + 0.00906i)16-s + 0.727·17-s + 0.825·19-s + (0.532 + 0.921i)20-s + (0.0152 − 0.0263i)22-s + (−0.295 + 0.512i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.173 - 0.984i$
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: \((2,\ 243,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11498 + 0.935584i\)
\(L(\frac12)\) \(\approx\) \(1.11498 + 0.935584i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.439 - 0.761i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.93 - 3.35i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.09 - 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0812 + 0.140i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.20 - 2.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 3.59T + 19T^{2} \)
23 \( 1 + (1.41 - 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.35 + 5.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.59 + 4.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 + (-2.90 + 5.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.11 - 5.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.69 + 6.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.40T + 53T^{2} \)
59 \( 1 + (-2.56 + 4.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.89 - 3.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.93 + 5.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 - 8.68T + 73T^{2} \)
79 \( 1 + (-0.634 - 1.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.23 - 7.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.72T + 89T^{2} \)
97 \( 1 + (-1.95 - 3.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.88429103009151146784136129190, −11.52451264923225209680545675864, −10.52583720286601080912038384257, −9.605438213099162090633668517345, −7.931719553792564996207976393848, −7.30019113503701205806035151723, −6.31836484524024178368009372907, −5.30142074578378646748794513024, −3.80975636278497073286062357222, −2.31587401168300181589833678732, 1.24422122659166849697688021546, 3.32859874121852764116813844782, 4.38081312457330384437325065946, 5.23591009169875842507582832343, 7.30047233105987494304834424223, 7.896874613833659631517435676703, 8.805147565833705476116494063703, 10.20023069886119179390523825482, 11.19367838957481517556784548568, 12.14010100021807217156725003483

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