Properties

Label 1-21e2-441.418-r1-0-0
Degree $1$
Conductor $441$
Sign $0.328 - 0.944i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (−0.900 − 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.988 − 0.149i)11-s + (0.988 + 0.149i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 + 0.680i)20-s + (−0.733 + 0.680i)22-s + (0.955 + 0.294i)23-s + (0.365 − 0.930i)25-s + (0.733 − 0.680i)26-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (−0.900 − 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.988 − 0.149i)11-s + (0.988 + 0.149i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 + 0.680i)20-s + (−0.733 + 0.680i)22-s + (0.955 + 0.294i)23-s + (0.365 − 0.930i)25-s + (0.733 − 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705760665 - 1.212586814i\)
\(L(\frac12)\) \(\approx\) \(1.705760665 - 1.212586814i\)
\(L(1)\) \(\approx\) \(1.149636112 - 0.4978208718i\)
\(L(1)\) \(\approx\) \(1.149636112 - 0.4978208718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (-0.0747 - 0.997i)T \)
43 \( 1 + (0.0747 - 0.997i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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

−23.85863876213601247304946939971, −23.29816570204150651609656990354, −22.66446527405125602447818419430, −21.442282444783317406747860975468, −20.71872239247545210343173938685, −19.992748246346017261137442503883, −18.557113579143438355872365277945, −18.004414250318209677617104634849, −16.57997618360412060958241506238, −16.239022855656085495709480700130, −15.33520631862871492437676722738, −14.61982001564420118413210111331, −13.18253026247843113167789694257, −13.02532674124944249923682855274, −11.75102967620529258177729130411, −11.03267398768833451066455799300, −9.409425094765460179047250933344, −8.4717058637216619818672645138, −7.67949387094106908569165985777, −6.888247368926278354934615235230, −5.46631129717539558159659157252, −4.91016145527678484718098957731, −3.72799691835984445569093758774, −2.837715572950596866327561445358, −0.78604760816716008739955925204, 0.67629612901400912159524212465, 2.08422224788111739353721649990, 3.37935122224275751366562845056, 3.837905554138360523153391213558, 5.24569066145385458940786588865, 6.08596968901945285949001512016, 7.38874594655055404165473212159, 8.36796369027830222028472896580, 9.64766963904016790434419620112, 10.72750962944158210641147861028, 11.14634628494430355974393290604, 12.21175779111107887061052344148, 13.01488641902709688165348500782, 13.964162430191205511847967712209, 14.886452455142450282977340521363, 15.57543172273831748710988592163, 16.51602369053459221604950448721, 18.10205501436170788698000642419, 18.75349882430632165722659470097, 19.29076497498693788029226378955, 20.48772080193805602891815171311, 20.9860413401487442351997616426, 21.98165736751309355177098854912, 22.92935443160089215191021090128, 23.4306368522188762855699929313

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