↑

Properties

Label 1-12e3-1728.1013-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.662 + 0.749i$
Analytic cond. $185.699$
Root an. cond. $185.699$
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  + (0.976 + 0.216i)5-s + (0.0871 + 0.996i)7-s + (0.537 − 0.843i)11-s + (0.0436 + 0.999i)13-s + (0.866 − 0.5i)17-s + (−0.991 − 0.130i)19-s + (0.0871 − 0.996i)23-s + (0.906 + 0.422i)25-s + (−0.737 + 0.675i)29-s + (−0.766 + 0.642i)31-s + (−0.130 + 0.991i)35-s + (−0.991 + 0.130i)37-s + (−0.906 + 0.422i)41-s + (0.537 − 0.843i)43-s + (−0.642 + 0.766i)47-s + ⋯
L(s)  = 1  + (0.976 + 0.216i)5-s + (0.0871 + 0.996i)7-s + (0.537 − 0.843i)11-s + (0.0436 + 0.999i)13-s + (0.866 − 0.5i)17-s + (−0.991 − 0.130i)19-s + (0.0871 − 0.996i)23-s + (0.906 + 0.422i)25-s + (−0.737 + 0.675i)29-s + (−0.766 + 0.642i)31-s + (−0.130 + 0.991i)35-s + (−0.991 + 0.130i)37-s + (−0.906 + 0.422i)41-s + (0.537 − 0.843i)43-s + (−0.642 + 0.766i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7661435658 + 1.699125563i\)
\(L(\frac12)\) \(\approx\) \(0.7661435658 + 1.699125563i\)
\(L(1)\) \(\approx\) \(1.176602736 + 0.2966252643i\)
\(L(1)\) \(\approx\) \(1.176602736 + 0.2966252643i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.976 + 0.216i)T \)
7 \( 1 + (0.0871 + 0.996i)T \)
11 \( 1 + (0.537 - 0.843i)T \)
13 \( 1 + (0.0436 + 0.999i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.991 - 0.130i)T \)
23 \( 1 + (0.0871 - 0.996i)T \)
29 \( 1 + (-0.737 + 0.675i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (-0.991 + 0.130i)T \)
41 \( 1 + (-0.906 + 0.422i)T \)
43 \( 1 + (0.537 - 0.843i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (0.976 + 0.216i)T \)
61 \( 1 + (0.953 + 0.300i)T \)
67 \( 1 + (-0.999 + 0.0436i)T \)
71 \( 1 + (-0.258 + 0.965i)T \)
73 \( 1 + (0.258 + 0.965i)T \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (0.675 + 0.737i)T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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

−19.863204640988261474374022516298, −19.25351476985948562734012868631, −18.14241638903294122925041267272, −17.46062718665142534787205428358, −17.06253406700458644354495440348, −16.40734436197181835803629786744, −15.117356588633353174114978765674, −14.71514789875005826642868167712, −13.784712267314907219167813502146, −13.102074295980920065480768822643, −12.58957363086949747515093059510, −11.53120825454752201412866965814, −10.511206325253757793703928690284, −10.07343177865949427550270579377, −9.382842015444166586492364710387, −8.35222395359022366988090330746, −7.52476055920030419358143396135, −6.76350836348869990730441502970, −5.825047338515898392889446286057, −5.161022835998775709249323431613, −4.100698571148583238172261823539, −3.395569354599394652623656654250, −2.0359792895474002494671599444, −1.441244097705308853177126743928, −0.31035306627408526130173705685, 1.21961786163919999117026606278, 2.059434251367943376099814252832, 2.85658436035348496671404969493, 3.83978731020577771365870886155, 5.05493761947874001472751055004, 5.67071897504820682606139643969, 6.47590063231207978809267923567, 7.08814811894567587860607508139, 8.6308829155225591453247669477, 8.80393893118048498744383106772, 9.6927033512798986660918734897, 10.57892655815349313523050822136, 11.35415847532905818322070065575, 12.14873554651992434901107330256, 12.882225726802567677340269274347, 13.80713428958126059405846348908, 14.46202235642806001410285614073, 14.90267662285974686591795696971, 16.189384063030355206652538639, 16.60716891671631969048086357690, 17.41598909226178328208784568095, 18.341889746621825982474134873788, 18.80506755726154963310264348353, 19.36814360500199530566428088034, 20.68514058402073279666787336714

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