↑

Properties

Label 1-459-459.115-r0-0-0
Degree $1$
Conductor $459$
Sign $0.780 + 0.625i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.342 − 0.939i)5-s + (0.984 − 0.173i)7-s + (0.5 + 0.866i)8-s + (0.866 + 0.5i)10-s + (−0.342 + 0.939i)11-s + (0.766 + 0.642i)13-s + (−0.642 + 0.766i)14-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)19-s + (−0.984 + 0.173i)20-s + (−0.342 − 0.939i)22-s + (−0.984 − 0.173i)23-s + (−0.766 + 0.642i)25-s − 26-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.342 − 0.939i)5-s + (0.984 − 0.173i)7-s + (0.5 + 0.866i)8-s + (0.866 + 0.5i)10-s + (−0.342 + 0.939i)11-s + (0.766 + 0.642i)13-s + (−0.642 + 0.766i)14-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)19-s + (−0.984 + 0.173i)20-s + (−0.342 − 0.939i)22-s + (−0.984 − 0.173i)23-s + (−0.766 + 0.642i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(2.13158\)
Root analytic conductor: \(2.13158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 459,\ (0:\ ),\ 0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9037174843 + 0.3172282382i\)
\(L(\frac12)\) \(\approx\) \(0.9037174843 + 0.3172282382i\)
\(L(1)\) \(\approx\) \(0.7975035586 + 0.1603043252i\)
\(L(1)\) \(\approx\) \(0.7975035586 + 0.1603043252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (0.984 - 0.173i)T \)
11 \( 1 + (-0.342 + 0.939i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (0.984 + 0.173i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.984 - 0.173i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.642 - 0.766i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.342 - 0.939i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.896937720714948876168792010462, −22.78694631590232082208503083738, −21.91678230453845285692677779073, −21.22384174580016615383305297481, −20.390053131951078998674427801347, −19.41573030645677926376768991995, −18.66911876654669784024255143933, −17.951181788990200402890415307741, −17.3907080085782295855216593334, −15.93580518710505417102114816200, −15.48453107103999282787550452078, −14.10035272408676302390737213353, −13.4008515300261795410355640005, −11.98735559185877278383479151432, −11.324280735641336463774012927156, −10.73177223161375180479800089033, −9.828584162619746151886654530570, −8.38011819665311234081565705645, −8.090058430159471639695196650934, −6.946342652351188206397379998246, −5.75334381304128658304164693227, −4.24224069831264978710935022631, −3.17653630061457121375958498886, −2.34328032538154737803554688752, −0.8703279029057145702003842785, 1.140981813085535355813510730256, 1.9814990275488780316690900542, 4.13253945352987597711034811519, 4.921767931306353598463148549928, 5.89125909759708478576382114075, 7.1862836737511203886536492205, 8.0109109545847719229304830166, 8.641591226118088999841283854846, 9.64622836695480234859464043640, 10.614655106529927727645199452404, 11.63120649952287325372668233079, 12.50499701908657633493497442698, 13.91884286189931522636587253912, 14.48696868123771905040114492150, 15.84767104194828981306901918742, 16.033988108432734369294333929354, 17.30598249395357356188257754195, 17.77251484998405909773205209501, 18.73292105184794160993568110571, 19.71064356203970488894426654560, 20.66746282416449535700178189545, 20.939326398504705740924148684458, 22.66196895877534510801160526060, 23.57386136093088327941217982362, 24.04390158221194967924572645998

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