Properties

Label 2-21e2-441.142-c1-0-48
Degree $2$
Conductor $441$
Sign $-0.840 - 0.542i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.880 − 2.24i)2-s + (−0.697 + 1.58i)3-s + (−2.79 − 2.58i)4-s + (−0.193 + 0.0931i)5-s + (2.94 + 2.96i)6-s + (−2.55 − 0.680i)7-s + (−3.92 + 1.88i)8-s + (−2.02 − 2.21i)9-s + (0.0386 + 0.515i)10-s + (−2.29 − 2.88i)11-s + (6.05 − 2.61i)12-s + (1.02 − 2.62i)13-s + (−3.77 + 5.13i)14-s + (−0.0127 − 0.371i)15-s + (0.215 + 2.87i)16-s + (−4.84 + 4.49i)17-s + ⋯
L(s)  = 1  + (0.622 − 1.58i)2-s + (−0.402 + 0.915i)3-s + (−1.39 − 1.29i)4-s + (−0.0864 + 0.0416i)5-s + (1.20 + 1.20i)6-s + (−0.966 − 0.257i)7-s + (−1.38 + 0.668i)8-s + (−0.675 − 0.737i)9-s + (0.0122 + 0.163i)10-s + (−0.692 − 0.868i)11-s + (1.74 − 0.755i)12-s + (0.285 − 0.727i)13-s + (−1.00 + 1.37i)14-s + (−0.00329 − 0.0959i)15-s + (0.0538 + 0.718i)16-s + (−1.17 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.840 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181951 + 0.617820i\)
\(L(\frac12)\) \(\approx\) \(0.181951 + 0.617820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.697 - 1.58i)T \)
7 \( 1 + (2.55 + 0.680i)T \)
good2 \( 1 + (-0.880 + 2.24i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (0.193 - 0.0931i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.29 + 2.88i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.02 + 2.62i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (4.84 - 4.49i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.18 - 3.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 5.82i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.39 + 1.35i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-3.77 + 6.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.89 - 1.50i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-2.47 - 1.68i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (2.32 - 1.58i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-4.35 + 11.0i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-8.19 + 2.52i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.335 - 0.228i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (1.92 - 1.78i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-1.73 + 3.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.75 + 12.0i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.79 + 1.02i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-6.35 - 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.74 + 7.00i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.422 - 1.07i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-2.51 + 4.35i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63612166962713538540459678101, −10.21245516170214566559841610736, −9.269279406061263510554321229458, −8.245525754100112349267993192151, −6.27390728778498238239097202010, −5.55115680636415266574660963123, −4.20838197160669604991904574623, −3.62921209759183876770315039804, −2.55613636556161161927042019058, −0.32896310686168311910551818440, 2.54005247900338231766103602471, 4.32456626015214210432766083427, 5.27446374823681020951490013119, 6.23062775578365467876432093366, 6.98048308798406694120610564943, 7.43081929785698077426036664588, 8.638425898354811930991295046742, 9.435452049426435943184615368434, 10.96301446813289480907893283743, 12.01643125791634477238920862452

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