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Properties

Label 2-21e2-49.22-c1-0-7
Degree $2$
Conductor $441$
Sign $0.474 - 0.880i$
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  + (1.60 − 0.772i)2-s + (0.728 − 0.913i)4-s + (−0.987 + 4.32i)5-s + (−2.63 − 0.271i)7-s + (−0.329 + 1.44i)8-s + (1.75 + 7.70i)10-s + (−0.951 + 0.458i)11-s + (4.24 − 2.04i)13-s + (−4.42 + 1.59i)14-s + (1.10 + 4.84i)16-s + (2.95 + 3.70i)17-s + 2.09·19-s + (3.23 + 4.05i)20-s + (−1.17 + 1.46i)22-s + (0.565 − 0.709i)23-s + ⋯
L(s)  = 1  + (1.13 − 0.546i)2-s + (0.364 − 0.456i)4-s + (−0.441 + 1.93i)5-s + (−0.994 − 0.102i)7-s + (−0.116 + 0.510i)8-s + (0.555 + 2.43i)10-s + (−0.286 + 0.138i)11-s + (1.17 − 0.567i)13-s + (−1.18 + 0.426i)14-s + (0.276 + 1.21i)16-s + (0.716 + 0.899i)17-s + 0.481·19-s + (0.722 + 0.906i)20-s + (−0.249 + 0.313i)22-s + (0.117 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67006 + 0.997570i\)
\(L(\frac12)\) \(\approx\) \(1.67006 + 0.997570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.63 + 0.271i)T \)
good2 \( 1 + (-1.60 + 0.772i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (0.987 - 4.32i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (0.951 - 0.458i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-4.24 + 2.04i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-2.95 - 3.70i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 - 2.09T + 19T^{2} \)
23 \( 1 + (-0.565 + 0.709i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (1.79 + 2.25i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 1.97T + 31T^{2} \)
37 \( 1 + (-0.285 - 0.357i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (0.917 - 4.02i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (1.67 + 7.35i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-7.33 + 3.53i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.774 + 0.971i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.293 + 1.28i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-7.12 - 8.93i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 2.75T + 67T^{2} \)
71 \( 1 + (-2.88 + 3.62i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.97 - 2.39i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 8.21T + 79T^{2} \)
83 \( 1 + (13.7 + 6.60i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (10.0 + 4.85i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 - 5.72T + 97T^{2} \)
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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

−11.33790827498873083247095924027, −10.56240254567097929154587094167, −10.05034922593377710489772860587, −8.396367285962724599433947597913, −7.37794152609589837654550462264, −6.33850548432399654929009273541, −5.69347771433978834085259516781, −3.84835701726605754044343707084, −3.43713200327377718253300111004, −2.52414779568531386185321086743, 0.906713363285009295994026964563, 3.39409441318178651305592473050, 4.25705065130931601498854053857, 5.25205891850822284586168009914, 5.86300678179157025289571946082, 7.05745077729639921963013934093, 8.188017365609202088074488203143, 9.192485498036603583168002431642, 9.711117939016645922588123905288, 11.39478610954345411480224257431

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