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Properties

Label 2-21e2-63.31-c2-0-56
Degree $2$
Conductor $441$
Sign $0.287 + 0.957i$
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  + (1.30 − 2.26i)2-s + (2.30 + 1.92i)3-s + (−1.41 − 2.45i)4-s − 3.68i·5-s + (7.36 − 2.70i)6-s + 3.03·8-s + (1.61 + 8.85i)9-s + (−8.34 − 4.81i)10-s + 13.2·11-s + (1.45 − 8.39i)12-s + (−16.2 − 9.37i)13-s + (7.08 − 8.48i)15-s + (9.64 − 16.7i)16-s + (7.13 + 4.11i)17-s + (22.1 + 7.92i)18-s + (19.6 − 11.3i)19-s + ⋯
L(s)  = 1  + (0.653 − 1.13i)2-s + (0.767 + 0.640i)3-s + (−0.354 − 0.614i)4-s − 0.737i·5-s + (1.22 − 0.450i)6-s + 0.379·8-s + (0.179 + 0.983i)9-s + (−0.834 − 0.481i)10-s + 1.20·11-s + (0.121 − 0.699i)12-s + (−1.24 − 0.721i)13-s + (0.472 − 0.565i)15-s + (0.603 − 1.04i)16-s + (0.419 + 0.242i)17-s + (1.23 + 0.440i)18-s + (1.03 − 0.595i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.77224 - 2.06300i\)
\(L(\frac12)\) \(\approx\) \(2.77224 - 2.06300i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.30 - 1.92i)T \)
7 \( 1 \)
good2 \( 1 + (-1.30 + 2.26i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + 3.68iT - 25T^{2} \)
11 \( 1 - 13.2T + 121T^{2} \)
13 \( 1 + (16.2 + 9.37i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-7.13 - 4.11i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-19.6 + 11.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + 0.738T + 529T^{2} \)
29 \( 1 + (6.13 + 10.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (28.6 - 16.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-4.98 - 8.63i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-23.3 - 13.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-10.1 - 17.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (69.3 + 40.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (19.5 - 33.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (40.3 - 23.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-34.7 - 20.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (54.5 + 94.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 83.8T + 5.04e3T^{2} \)
73 \( 1 + (-32.3 - 18.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (49.4 - 85.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (124. - 71.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (115. - 66.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-58.0 + 33.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.81281721436597479407269593319, −9.774232951025464187755436194760, −9.343077786186679723286738513530, −8.155435920794960885633553090834, −7.21425136256847610776395769936, −5.31132526222535147889656801016, −4.63338679513705114613631393458, −3.62394898130423756191871554556, −2.71032949320689420530947137210, −1.33153798270530670689986209413, 1.68117932967778931644175393864, 3.21073943661362339572508787069, 4.29662066978275936220099006027, 5.64458586381933459859281113335, 6.70556624458837507545603214563, 7.16196796847460515715852879600, 7.88114431201244992050786444837, 9.164570179864480159687554841660, 9.897727652403483064466049956902, 11.31867724664815001428531433822

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