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Properties

Label 2-3e5-9.7-c1-0-6
Degree $2$
Conductor $243$
Sign $0.939 + 0.342i$
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  + (0.673 − 1.16i)2-s + (0.0923 + 0.160i)4-s + (0.826 + 1.43i)5-s + (−1.20 + 2.08i)7-s + 2.94·8-s + 2.22·10-s + (2.97 − 5.14i)11-s + (1.61 + 2.79i)13-s + (1.62 + 2.81i)14-s + (1.79 − 3.11i)16-s − 3·17-s − 6.63·19-s + (−0.152 + 0.264i)20-s + (−4.00 − 6.93i)22-s + (−1.47 − 2.54i)23-s + ⋯
L(s)  = 1  + (0.476 − 0.825i)2-s + (0.0461 + 0.0800i)4-s + (0.369 + 0.640i)5-s + (−0.455 + 0.789i)7-s + 1.04·8-s + 0.704·10-s + (0.896 − 1.55i)11-s + (0.447 + 0.775i)13-s + (0.434 + 0.751i)14-s + (0.449 − 0.778i)16-s − 0.727·17-s − 1.52·19-s + (−0.0341 + 0.0591i)20-s + (−0.853 − 1.47i)22-s + (−0.306 − 0.531i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73238 - 0.305466i\)
\(L(\frac12)\) \(\approx\) \(1.73238 - 0.305466i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.673 + 1.16i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.826 - 1.43i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.20 - 2.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.97 + 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.61 - 2.79i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 6.63T + 19T^{2} \)
23 \( 1 + (1.47 + 2.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.645 - 1.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.294 - 0.509i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.0418T + 37T^{2} \)
41 \( 1 + (2.45 + 4.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.59 + 4.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.86 - 3.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (3.67 + 6.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.52 - 9.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.928 + 1.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 - 5.55T + 73T^{2} \)
79 \( 1 + (-1.89 + 3.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.99 + 3.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.15T + 89T^{2} \)
97 \( 1 + (0.130 - 0.225i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.02283837215229683656125345832, −11.10446370833251009214971519335, −10.61532264713517661323904206874, −9.153040500409752154888557604260, −8.405089211171503239844124680947, −6.66161844990791583052990623292, −6.11211145345343714273002250432, −4.30572697148041977933841722946, −3.21330485047285602568872488232, −2.09821743474737960423295169557, 1.66853619473493056193113358103, 4.06257227749252560016882852512, 4.88939337116370981639806746915, 6.22629403480118035471323786931, 6.88557631982669310300879027552, 7.972916404870671000216071273160, 9.314941455074169989083533308641, 10.16563213373329202973087494875, 11.10447545673076465463757130078, 12.62460816722953855594019707110

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