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Properties

Label 2-21e2-63.13-c2-0-50
Degree $2$
Conductor $441$
Sign $0.940 - 0.339i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.902 + 1.56i)2-s + (2.28 − 1.94i)3-s + (0.371 − 0.643i)4-s + (4.98 + 2.87i)5-s + (5.09 + 1.81i)6-s + 8.56·8-s + (1.44 − 8.88i)9-s + 10.3i·10-s + (5.09 + 8.82i)11-s + (−0.401 − 2.19i)12-s + (−9.76 − 5.63i)13-s + (16.9 − 3.11i)15-s + (6.23 + 10.8i)16-s + 17.7i·17-s + (15.1 − 5.75i)18-s − 30.7i·19-s + ⋯
L(s)  = 1  + (0.451 + 0.781i)2-s + (0.761 − 0.647i)3-s + (0.0929 − 0.160i)4-s + (0.996 + 0.575i)5-s + (0.849 + 0.302i)6-s + 1.07·8-s + (0.160 − 0.987i)9-s + 1.03i·10-s + (0.463 + 0.802i)11-s + (−0.0334 − 0.182i)12-s + (−0.750 − 0.433i)13-s + (1.13 − 0.207i)15-s + (0.389 + 0.675i)16-s + 1.04i·17-s + (0.843 − 0.319i)18-s − 1.61i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.940 - 0.339i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.60389 + 0.631153i\)
\(L(\frac12)\) \(\approx\) \(3.60389 + 0.631153i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.28 + 1.94i)T \)
7 \( 1 \)
good2 \( 1 + (-0.902 - 1.56i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-4.98 - 2.87i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.09 - 8.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.76 + 5.63i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 17.7iT - 289T^{2} \)
19 \( 1 + 30.7iT - 361T^{2} \)
23 \( 1 + (15.6 - 27.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-1.48 - 2.56i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (25.0 + 14.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 22.1T + 1.36e3T^{2} \)
41 \( 1 + (9.95 + 5.74i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-19.6 - 34.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-18.5 + 10.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 45.7T + 2.80e3T^{2} \)
59 \( 1 + (11.1 + 6.42i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (72.4 - 41.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (10.1 - 17.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 92.6T + 5.04e3T^{2} \)
73 \( 1 + 102. iT - 5.32e3T^{2} \)
79 \( 1 + (-50.2 - 86.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (20.6 - 11.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 55.9iT - 7.92e3T^{2} \)
97 \( 1 + (69.6 - 40.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.81445066233557405504981959314, −9.878753402027952652118937930064, −9.223847258314558447721006108984, −7.80738759438550230511678773928, −7.13093411518222286244029726893, −6.38372832528349539492302656506, −5.55091990639041942216534537708, −4.19650909050858453762268110266, −2.57237446596611253480646038612, −1.62433090957237934379620413627, 1.69031134677528799233598273333, 2.67014260156225410242237071249, 3.81461364961463862361841066010, 4.74076466837563007774075199459, 5.81718979087745998081801550709, 7.31888993481147057915910996687, 8.360621405822654767526259570727, 9.252077005927362702250456598758, 10.00831404195258627075516953429, 10.75079640648091928157835954048

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