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Properties

Label 1-21e2-441.221-r1-0-0
Degree $1$
Conductor $441$
Sign $0.0924 + 0.995i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.222 + 0.974i)5-s + (0.222 + 0.974i)8-s + (0.955 − 0.294i)10-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.222 + 0.974i)5-s + (0.222 + 0.974i)8-s + (0.955 − 0.294i)10-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0924 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0924 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0924 + 0.995i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.0924 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9041407249 + 0.8240597576i\)
\(L(\frac12)\) \(\approx\) \(0.9041407249 + 0.8240597576i\)
\(L(1)\) \(\approx\) \(0.9402475654 - 0.05932401980i\)
\(L(1)\) \(\approx\) \(0.9402475654 - 0.05932401980i\)

Euler product

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

−23.89113955224504643336104546175, −22.98688528867698995003450384071, −22.02942602824605415255195977820, −21.279953185263565619946535798147, −20.04515290327985001334324791298, −19.369148188254027630972122844732, −18.1343601702591645113454329308, −17.430091971371441241350587250905, −16.62569903621477362545149018535, −16.01743258105365513667695872696, −14.95732019190054288741930798842, −14.1406774526489508822586138765, −13.144707380327714234437349373675, −12.546221690167154757883770101553, −11.2446106617354642402516023645, −9.8296896850681429939635869418, −9.19004220031118747287402528425, −8.26349404227016890224324417454, −7.47010879831329853542806608752, −6.139098978338522469682820723088, −5.50004246327120542217404892036, −4.479658985654528054182601527495, −3.39907296943967779839377995514, −1.39798320963257561210676178530, −0.35580085411054145286765578949, 1.45342295362495438847734874221, 2.31523900960791021905924274112, 3.57365640014800014556947893643, 4.271227315389967519524885712335, 5.76016948575842740785182571853, 6.77959541291206151849594622445, 7.965232410973739128159188390999, 9.11194553323068182203207801076, 9.98660013647708322760540160138, 10.64236815815089900597194070238, 11.77412443975813842013663212737, 12.24011481177913498962386491438, 13.5843715193238620885853348156, 14.344400756043114662577699257554, 14.88621753017455379174455558624, 16.559694126823541583208304263982, 17.270165879938586227845721504, 18.36888242223027725198944370859, 18.86763518242597343857563138699, 19.65364991208581518847243688987, 20.70230759103889414703550612095, 21.48159690587335743017760147802, 22.21621699516735433801896662654, 22.94653951402684885982040032282, 23.69488785610262311219711361846

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