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Properties

Label 2-21e2-147.104-c1-0-2
Degree $2$
Conductor $441$
Sign $0.804 - 0.594i$
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  + (−2.12 − 1.69i)2-s + (1.19 + 5.25i)4-s + (2.70 + 1.30i)5-s + (−0.967 + 2.46i)7-s + (3.99 − 8.30i)8-s + (−3.53 − 7.34i)10-s + (−1.87 − 1.49i)11-s + (1.93 + 1.54i)13-s + (6.23 − 3.59i)14-s + (−12.8 + 6.19i)16-s + (−1.15 + 5.04i)17-s + 0.270i·19-s + (−3.59 + 15.7i)20-s + (1.44 + 6.34i)22-s + (−8.11 + 1.85i)23-s + ⋯
L(s)  = 1  + (−1.50 − 1.19i)2-s + (0.599 + 2.62i)4-s + (1.20 + 0.582i)5-s + (−0.365 + 0.930i)7-s + (1.41 − 2.93i)8-s + (−1.11 − 2.32i)10-s + (−0.564 − 0.449i)11-s + (0.536 + 0.427i)13-s + (1.66 − 0.960i)14-s + (−3.21 + 1.54i)16-s + (−0.279 + 1.22i)17-s + 0.0621i·19-s + (−0.804 + 3.52i)20-s + (0.308 + 1.35i)22-s + (−1.69 + 0.386i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.804 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619376 + 0.203889i\)
\(L(\frac12)\) \(\approx\) \(0.619376 + 0.203889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.967 - 2.46i)T \)
good2 \( 1 + (2.12 + 1.69i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.70 - 1.30i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.87 + 1.49i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.93 - 1.54i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.15 - 5.04i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 0.270iT - 19T^{2} \)
23 \( 1 + (8.11 - 1.85i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.64 - 0.603i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 8.55iT - 31T^{2} \)
37 \( 1 + (-1.31 + 5.77i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (0.862 + 0.415i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-5.37 + 2.58i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.40 - 6.77i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.39 + 0.317i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-5.63 + 2.71i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-12.1 - 2.76i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.18T + 67T^{2} \)
71 \( 1 + (-3.43 + 0.784i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.58 - 5.25i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 8.56T + 79T^{2} \)
83 \( 1 + (-0.701 - 0.879i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-10.3 - 13.0i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−10.82693617892110616144817646288, −10.31564337331722554118493554916, −9.559496181072841192981623109400, −8.762214667022326912757102166112, −8.073798363229224763575354350325, −6.67093845598685873165346217592, −5.83027713709406512657839092499, −3.68522087111118955208751341276, −2.54696209800427071935030891257, −1.76327345783077534725960579528, 0.65781295383556089492069352501, 2.12972544261797417743684373093, 4.71596073221762955433209751769, 5.77475522111710406290695145568, 6.47075853456269811336328995918, 7.48469946500641960875563420779, 8.252102641074564843588765921411, 9.270740384475159547822140139397, 9.995512102199861653667932388325, 10.27453411157929852470378552178

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