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Properties

Label 1-3e5-243.169-r0-0-0
Degree $1$
Conductor $243$
Sign $0.0258 + 0.999i$
Analytic cond. $1.12848$
Root an. cond. $1.12848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.813 + 0.581i)2-s + (0.323 + 0.946i)4-s + (−0.135 + 0.990i)5-s + (0.856 − 0.516i)7-s + (−0.286 + 0.957i)8-s + (−0.686 + 0.727i)10-s + (0.952 − 0.305i)11-s + (0.249 − 0.968i)13-s + (0.996 + 0.0774i)14-s + (−0.790 + 0.612i)16-s + (−0.0581 + 0.998i)17-s + (−0.835 + 0.549i)19-s + (−0.981 + 0.192i)20-s + (0.952 + 0.305i)22-s + (−0.875 + 0.483i)23-s + ⋯
L(s)  = 1  + (0.813 + 0.581i)2-s + (0.323 + 0.946i)4-s + (−0.135 + 0.990i)5-s + (0.856 − 0.516i)7-s + (−0.286 + 0.957i)8-s + (−0.686 + 0.727i)10-s + (0.952 − 0.305i)11-s + (0.249 − 0.968i)13-s + (0.996 + 0.0774i)14-s + (−0.790 + 0.612i)16-s + (−0.0581 + 0.998i)17-s + (−0.835 + 0.549i)19-s + (−0.981 + 0.192i)20-s + (0.952 + 0.305i)22-s + (−0.875 + 0.483i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.0258 + 0.999i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.0258 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433942477 + 1.397336628i\)
\(L(\frac12)\) \(\approx\) \(1.433942477 + 1.397336628i\)
\(L(1)\) \(\approx\) \(1.468002084 + 0.8327310743i\)
\(L(1)\) \(\approx\) \(1.468002084 + 0.8327310743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.813 + 0.581i)T \)
5 \( 1 + (-0.135 + 0.990i)T \)
7 \( 1 + (0.856 - 0.516i)T \)
11 \( 1 + (0.952 - 0.305i)T \)
13 \( 1 + (0.249 - 0.968i)T \)
17 \( 1 + (-0.0581 + 0.998i)T \)
19 \( 1 + (-0.835 + 0.549i)T \)
23 \( 1 + (-0.875 + 0.483i)T \)
29 \( 1 + (-0.565 - 0.824i)T \)
31 \( 1 + (0.987 - 0.154i)T \)
37 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (0.0968 + 0.995i)T \)
43 \( 1 + (0.533 - 0.845i)T \)
47 \( 1 + (0.987 + 0.154i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.211 - 0.977i)T \)
61 \( 1 + (0.323 - 0.946i)T \)
67 \( 1 + (-0.565 + 0.824i)T \)
71 \( 1 + (0.973 + 0.230i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (-0.910 + 0.413i)T \)
83 \( 1 + (0.0968 - 0.995i)T \)
89 \( 1 + (0.973 - 0.230i)T \)
97 \( 1 + (-0.135 - 0.990i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.62929881249506015906001445673, −24.4794658739726654888583293749, −24.24802000511057655158866379183, −23.14953390426731734985323285858, −22.08964087268362996417785101187, −21.228963121500400761971746359832, −20.555490004257220419806890666544, −19.68634614262702591222345013698, −18.71459115432374772885366094947, −17.51950251121135157935741182851, −16.356981744992293717086150109377, −15.40623439112982381025621264210, −14.34350291394182101819775303154, −13.62467178990323289639797501903, −12.24139775063509065226966281889, −11.91728996281129762327035072733, −10.8533811769869232948240512714, −9.35214614673776813902085193208, −8.71320770509567716526006449535, −7.0528254544278003622548926843, −5.80669869338751040354514985092, −4.68301133053385960057498071642, −4.10722485465271745217292950344, −2.3088985071064142250298499580, −1.30093845360344445458675271770, 1.986668969231054546990682467838, 3.52267380312379903950200153518, 4.18948360952352232990097265362, 5.75294710841683670727978696397, 6.52985238203070914218674811410, 7.71887267947496753743778762589, 8.38423521667686622524798398289, 10.273897123814226539736424382246, 11.182796918820381945888046589907, 12.06168403519443730918699846042, 13.358732851797336818974480855665, 14.25659611711725692702627029203, 14.88081893319522397021769014805, 15.72535652003363395774846079749, 17.199981840363980444310989672182, 17.52389770848050011694151289893, 18.92331149797147983458059191616, 20.05374902679373329674308352323, 21.090425558505043180193234073809, 21.960955979871647388813799467620, 22.759037379139675637267552220128, 23.542896760876489516467553664698, 24.43340064128907344601079865677, 25.36358573627428796362663910900, 26.24255746256227290723280149875

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