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Properties

Label 1-21e2-441.272-r0-0-0
Degree $1$
Conductor $441$
Sign $0.999 + 0.0142i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.826 − 0.563i)5-s + (0.900 + 0.433i)8-s + (0.900 − 0.433i)10-s + (0.988 + 0.149i)11-s + (−0.365 − 0.930i)13-s + (0.826 + 0.563i)16-s + (−0.222 + 0.974i)17-s − 19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.826 − 0.563i)5-s + (0.900 + 0.433i)8-s + (0.900 − 0.433i)10-s + (0.988 + 0.149i)11-s + (−0.365 − 0.930i)13-s + (0.826 + 0.563i)16-s + (−0.222 + 0.974i)17-s − 19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0142i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.999 + 0.0142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.923998086 + 0.02083030907i\)
\(L(\frac12)\) \(\approx\) \(2.923998086 + 0.02083030907i\)
\(L(1)\) \(\approx\) \(2.156679218 + 0.03019759526i\)
\(L(1)\) \(\approx\) \(2.156679218 + 0.03019759526i\)

Euler product

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

−24.137727857031038604927008592965, −22.912391770718900351422154901305, −22.395573092855550131884222731453, −21.4939212980851935188806815081, −21.05016471832816406018295341501, −19.7839812156713508474396163158, −19.16786728639155301205733626084, −18.014513662135685239737169748755, −16.9818665626618267194350865663, −16.19648573032904323493423210452, −15.02137000089346396829847238419, −14.215614209828210669434216549230, −13.81241738422977658565461960661, −12.692280500007263776735589823911, −11.716226231253165004757145833550, −10.99316344950427682708501900899, −9.91173547887421797074616687153, −9.0721446521066526635634571039, −7.37599182369069006192510660926, −6.57184184092693106920584968125, −5.8672582086343799265427740059, −4.66068281026875340855162600006, −3.70480773807578877400668348565, −2.470666697866732964475400578540, −1.663593427375952781692482506758, 1.51568788839376820962720775862, 2.47371433617371689821463641292, 3.87797142625185906181955045548, 4.695902076773280987356927037296, 5.898542653789488649146184312939, 6.34333287010488001261946511986, 7.71101585602416644848013443557, 8.717743845647541646738531633686, 9.94145256770593084325914326166, 10.81613637535391552130558124667, 12.041117872740705927590955336691, 12.756432868635157431354138702317, 13.42596724427583744299719554483, 14.499015383854207745869408790088, 15.052665879178990927507492582416, 16.22684441058696419668376508680, 17.13208363511321397186759304799, 17.56060073592040904860589717562, 19.16168668805591114060363769753, 20.09982379110120959950081428627, 20.6917328017319980973568155726, 21.74761498502629331008360591076, 22.19040264592696302829694363487, 23.119728936605724048084629698085, 24.36229747745158744614674789255

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