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Properties

Label 1-459-459.266-r0-0-0
Degree $1$
Conductor $459$
Sign $0.721 + 0.692i$
Analytic cond. $2.13158$
Root an. cond. $2.13158$
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  + (−0.0871 − 0.996i)2-s + (−0.984 + 0.173i)4-s + (0.0436 + 0.999i)5-s + (−0.843 − 0.537i)7-s + (0.258 + 0.965i)8-s + (0.991 − 0.130i)10-s + (0.999 + 0.0436i)11-s + (−0.642 − 0.766i)13-s + (−0.461 + 0.887i)14-s + (0.939 − 0.342i)16-s + (−0.965 + 0.258i)19-s + (−0.216 − 0.976i)20-s + (−0.0436 − 0.999i)22-s + (−0.216 + 0.976i)23-s + (−0.996 + 0.0871i)25-s + (−0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.0871 − 0.996i)2-s + (−0.984 + 0.173i)4-s + (0.0436 + 0.999i)5-s + (−0.843 − 0.537i)7-s + (0.258 + 0.965i)8-s + (0.991 − 0.130i)10-s + (0.999 + 0.0436i)11-s + (−0.642 − 0.766i)13-s + (−0.461 + 0.887i)14-s + (0.939 − 0.342i)16-s + (−0.965 + 0.258i)19-s + (−0.216 − 0.976i)20-s + (−0.0436 − 0.999i)22-s + (−0.216 + 0.976i)23-s + (−0.996 + 0.0871i)25-s + (−0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(2.13158\)
Root analytic conductor: \(2.13158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 459,\ (0:\ ),\ 0.721 + 0.692i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6577640402 + 0.2646725461i\)
\(L(\frac12)\) \(\approx\) \(0.6577640402 + 0.2646725461i\)
\(L(1)\) \(\approx\) \(0.7567682068 - 0.1369583807i\)
\(L(1)\) \(\approx\) \(0.7567682068 - 0.1369583807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.0871 - 0.996i)T \)
5 \( 1 + (0.0436 + 0.999i)T \)
7 \( 1 + (-0.843 - 0.537i)T \)
11 \( 1 + (0.999 + 0.0436i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.216 + 0.976i)T \)
29 \( 1 + (-0.300 + 0.953i)T \)
31 \( 1 + (0.537 + 0.843i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (0.300 + 0.953i)T \)
43 \( 1 + (-0.422 + 0.906i)T \)
47 \( 1 + (0.984 + 0.173i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.906 + 0.422i)T \)
61 \( 1 + (0.843 + 0.537i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.793 - 0.608i)T \)
73 \( 1 + (-0.991 - 0.130i)T \)
79 \( 1 + (0.461 + 0.887i)T \)
83 \( 1 + (0.0871 + 0.996i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.675 + 0.737i)T \)
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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

−24.05221054398656187671523677629, −23.1301086220962344246953317452, −22.178118330054934913636815844362, −21.60725126250224439007941204863, −20.31224072409291221407415422214, −19.24091861904901390696209453226, −18.83512311287477997217397113438, −17.324548328296475264198224982639, −16.93693874519154091715625270850, −16.14843667901978154915950686990, −15.2994673859057310061587912150, −14.38780708513232958762505600624, −13.4116846616231675671102086149, −12.57572920316753545690624081218, −11.85044293377180280934555957152, −10.13472546557823800142841006547, −9.14755346032597091331864664405, −8.880034130920148152877969777342, −7.63545596992497822845035541903, −6.46976572299541982875198661733, −5.90147979009379988289615578434, −4.60963357148880413201330302840, −3.96577124046240566038823644288, −2.13525578708580673816257846982, −0.43291411423207716600581347124, 1.37128051461053534482613795604, 2.76587653517132745371732628430, 3.45181371417462252146394844891, 4.42977271279896510365982557317, 5.91748159834664228732495898751, 6.92187394320985782972373679851, 7.96377633645278008902008574856, 9.28775935690946821984234296694, 10.02009620668389144442178873890, 10.698286302443709742248135766612, 11.64201152591002303764871864780, 12.586776078017143056978870357846, 13.42930870950837474811215885791, 14.36373150063100020340112076250, 15.06377900931209424489433897957, 16.51248224466321879524229696737, 17.41246473984047209910835390404, 18.10094115869762588273991798285, 19.3955399393431715473418862725, 19.437928523837569080420028272622, 20.456109539669018466055431939123, 21.74959787920858667180714012474, 22.13054009498060583990843634451, 22.973390958731021212391035277918, 23.57087400848931021587828205335

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