Properties

Label 1-21e2-441.292-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.427 + 0.904i$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.222 + 0.974i)8-s + (−0.955 − 0.294i)10-s + (0.826 + 0.563i)11-s + (−0.826 − 0.563i)13-s + (−0.222 − 0.974i)16-s + (0.988 − 0.149i)17-s + (0.5 + 0.866i)19-s + (0.988 − 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.222 + 0.974i)8-s + (−0.955 − 0.294i)10-s + (0.826 + 0.563i)11-s + (−0.826 − 0.563i)13-s + (−0.222 − 0.974i)16-s + (0.988 − 0.149i)17-s + (0.5 + 0.866i)19-s + (0.988 − 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.427 + 0.904i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.427 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7439247998 + 1.174640996i\)
\(L(\frac12)\) \(\approx\) \(0.7439247998 + 1.174640996i\)
\(L(1)\) \(\approx\) \(0.7912940546 + 0.3528785410i\)
\(L(1)\) \(\approx\) \(0.7912940546 + 0.3528785410i\)

Euler product

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

−23.99877597419965191309237809861, −22.21693324416311750904677593754, −21.8110137375300586337233981748, −20.80077391057541702326043723557, −20.168359534809999660692246712765, −19.20312705624305312795456113950, −18.502955030408911923355791541357, −17.35338989120180013417887749692, −16.84806304242071288239325236725, −16.23564751997951123302574246489, −14.8469926987540406559267714493, −13.825763010335866391424189634788, −12.76469778089984600103551744207, −12.00579704251742733209865050612, −11.09684267728123460896080378726, −9.95275979253405692422891685597, −9.278583780642095312605849814507, −8.598823613698685237728545658088, −7.39812618901562972667955318083, −6.430967195153587215262688147738, −5.239155575734119306844428910117, −3.91991041515304563976546651895, −2.64141904528819469191085751171, −1.55310429376998921264000296497, −0.53590095557196448651264508645, 1.21537497455728830964926026936, 2.21156214175028145779747190775, 3.47651079058731693714748692824, 5.31653601248257260731580650128, 5.94062057407851337127858092790, 7.22453172101273887625942902319, 7.58557324936445829997529796412, 9.1677273987090204070201998746, 9.71679270384396846227666751923, 10.49423005142310583526109078497, 11.52698857141112722710177389831, 12.56073872565645486267921611984, 14.02384690833502358692906564002, 14.63600882787879808399100998093, 15.33344006088205721711719874279, 16.683430084548540444556065502235, 17.16312975500595430819561353365, 18.084844085609496274422543722543, 18.7201460427235784146355783458, 19.697729326401443242518118209373, 20.47378859694276143359158387942, 21.53160565211670919106543272022, 22.5316155453839183505178611579, 23.257799943208163420505663310981, 24.482203629539939111545433305657

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