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Properties

Label 1-21e2-441.23-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

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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: $\chi_{441} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5444581692 + 0.7658959488i\)
\(L(\frac12)\) \(\approx\) \(0.5444581692 + 0.7658959488i\)
\(L(1)\) \(\approx\) \(0.6210784230 + 0.2907256368i\)
\(L(1)\) \(\approx\) \(0.6210784230 + 0.2907256368i\)

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 \( 1 + (-0.5 + 0.866i)T \)
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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

−23.41809056031479302342619363416, −22.64498386892171074040761963735, −21.69162946627516288634377600348, −20.87090111585239096193460413278, −19.924713010660725072938615810909, −19.4535873765547419977827349474, −18.665811750003348855294056266881, −17.52315077306180787634334560037, −16.71045165067042423632669694712, −16.150144606032323830558859570426, −14.83319406197866308111220832679, −13.82060830591485513935444163368, −12.59516569802071476126259950570, −11.927551367288920925531969009849, −11.44408344150161474474577577336, −10.00954976831910639691361691039, −9.45134478138627816746509899292, −8.23410164150252183947707698147, −7.746568630107678887111939960720, −6.463688518938881018628187479393, −4.782303505012599709680824275, −4.00833022299116397234187380337, −2.93864611588514243231974319744, −1.54787598213422644805465202017, −0.45469543210596151611211241919, 0.81427740693462716499513521354, 2.36653791705368807952067095654, 3.894405345202230936312976906068, 4.85704089814403629774708804686, 6.22909142314614415880390265355, 6.949269334613775928573355807547, 7.83967643486089690962306217750, 8.66574957447646580767973679875, 9.79816506930515479851468322214, 10.57149954213229688569967207945, 11.63162858788019997706695637496, 12.54360663490643809591770479854, 14.109778527056400051458161423135, 14.64266962766849493490757098242, 15.41342685092159951717619541222, 16.29723866228655275770089921595, 17.233144580935789090955300233220, 17.88294023746279287084372214629, 19.14428291319311239885215213283, 19.41220816907418123074433433523, 20.28190062927844821027188794132, 21.90032039202745250018025339729, 22.4877407964652578042531777832, 23.52294496753151216392720777641, 24.060197149088397788501954548494

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