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Properties

Label 1-459-459.130-r1-0-0
Degree $1$
Conductor $459$
Sign $0.970 + 0.239i$
Analytic cond. $49.3264$
Root an. cond. $49.3264$
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  + (0.819 − 0.573i)2-s + (0.342 − 0.939i)4-s + (0.887 + 0.461i)5-s + (−0.0436 + 0.999i)7-s + (−0.258 − 0.965i)8-s + (0.991 − 0.130i)10-s + (0.461 + 0.887i)11-s + (0.984 − 0.173i)13-s + (0.537 + 0.843i)14-s + (−0.766 − 0.642i)16-s + (−0.965 + 0.258i)19-s + (0.737 − 0.675i)20-s + (0.887 + 0.461i)22-s + (0.737 + 0.675i)23-s + (0.573 + 0.819i)25-s + (0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (0.819 − 0.573i)2-s + (0.342 − 0.939i)4-s + (0.887 + 0.461i)5-s + (−0.0436 + 0.999i)7-s + (−0.258 − 0.965i)8-s + (0.991 − 0.130i)10-s + (0.461 + 0.887i)11-s + (0.984 − 0.173i)13-s + (0.537 + 0.843i)14-s + (−0.766 − 0.642i)16-s + (−0.965 + 0.258i)19-s + (0.737 − 0.675i)20-s + (0.887 + 0.461i)22-s + (0.737 + 0.675i)23-s + (0.573 + 0.819i)25-s + (0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(49.3264\)
Root analytic conductor: \(49.3264\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 459,\ (1:\ ),\ 0.970 + 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.046628719 + 0.4910558444i\)
\(L(\frac12)\) \(\approx\) \(4.046628719 + 0.4910558444i\)
\(L(1)\) \(\approx\) \(2.045254556 - 0.1804704957i\)
\(L(1)\) \(\approx\) \(2.045254556 - 0.1804704957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.819 - 0.573i)T \)
5 \( 1 + (0.887 + 0.461i)T \)
7 \( 1 + (-0.0436 + 0.999i)T \)
11 \( 1 + (0.461 + 0.887i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.737 + 0.675i)T \)
29 \( 1 + (-0.976 + 0.216i)T \)
31 \( 1 + (-0.999 + 0.0436i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (0.976 + 0.216i)T \)
43 \( 1 + (0.996 - 0.0871i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.0871 + 0.996i)T \)
61 \( 1 + (0.0436 - 0.999i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.793 + 0.608i)T \)
73 \( 1 + (-0.991 - 0.130i)T \)
79 \( 1 + (0.537 - 0.843i)T \)
83 \( 1 + (-0.819 + 0.573i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.300 - 0.953i)T \)
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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

−23.76810905073397530937876180714, −22.92372142265347991200904942190, −22.02455473559594441467835065301, −21.1394649619829640677063903155, −20.68003640982813494959662794729, −19.62611417390942851934220777501, −18.32322061577977799774233325842, −17.269693986858172410757742006346, −16.68617686306339953214157639976, −16.11647389985705957039518654955, −14.72916577193288289088458859104, −14.07054249168745315553139845798, −13.21172455682616138408407958627, −12.82417343217175668784475894113, −11.30408550027118072640447549384, −10.68167070680779338562073496265, −9.16502525777453775262968848338, −8.47160618614131318614790347897, −7.21647703500962335993776048324, −6.29674612106406530300236918592, −5.61337380575339486722032916338, −4.36523387672018365109785049419, −3.634971625854650797513023280275, −2.22495082917664409670268389268, −0.81159338976933928545556052139, 1.429403654019517000860241279108, 2.23122833484638759677408794691, 3.244792533131133820199006121209, 4.42768629494461581051423356519, 5.672644514704744051115670109, 6.12136445113523525228252127767, 7.240546996015517068722028555207, 8.99453815650934858065238215851, 9.57884760281136244343808698032, 10.72022509145637034124941744105, 11.37913906233852091362199797999, 12.61319574465326646787440554977, 13.04768145255332337321077004003, 14.195322569732107624864976871284, 14.9143345787360317437750848138, 15.544152461475761902816479310951, 16.875154786818034992791605141090, 18.04773778828877383373360132141, 18.62059116068049746459356669485, 19.54304467820712150533039761956, 20.667192788671006965883128795069, 21.24888899453002875248713817438, 22.044858398798486042464633506477, 22.69456265330985385217265332017, 23.498260413564587394137812585098

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