Properties

Label 2-21e2-63.2-c2-0-40
Degree $2$
Conductor $441$
Sign $0.734 - 0.678i$
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  + (1.5 − 0.866i)2-s + 3·3-s + (−0.5 + 0.866i)4-s + 3.46i·5-s + (4.5 − 2.59i)6-s + 8.66i·8-s + 9·9-s + (2.99 + 5.19i)10-s + 1.73i·11-s + (−1.5 + 2.59i)12-s + (2 + 3.46i)13-s + 10.3i·15-s + (5.5 + 9.52i)16-s + (−13.5 + 7.79i)17-s + (13.5 − 7.79i)18-s + (−5.5 + 9.52i)19-s + ⋯
L(s)  = 1  + (0.750 − 0.433i)2-s + 3-s + (−0.125 + 0.216i)4-s + 0.692i·5-s + (0.750 − 0.433i)6-s + 1.08i·8-s + 9-s + (0.299 + 0.519i)10-s + 0.157i·11-s + (−0.125 + 0.216i)12-s + (0.153 + 0.266i)13-s + 0.692i·15-s + (0.343 + 0.595i)16-s + (−0.794 + 0.458i)17-s + (0.750 − 0.433i)18-s + (−0.289 + 0.501i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.734 - 0.678i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.08144 + 1.20567i\)
\(L(\frac12)\) \(\approx\) \(3.08144 + 1.20567i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
7 \( 1 \)
good2 \( 1 + (-1.5 + 0.866i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 3.46iT - 25T^{2} \)
11 \( 1 - 1.73iT - 121T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (13.5 - 7.79i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (5.5 - 9.52i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + 27.7iT - 529T^{2} \)
29 \( 1 + (-39 - 22.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (16 - 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-17 + 29.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (10.5 - 6.06i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 + 52.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-42 + 24.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (43.5 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (28 + 48.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 + (32.5 + 56.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (19 + 32.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (42 + 24.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-108 - 62.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-57.5 + 99.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.88412713466538810920506194542, −10.41390046773913804259601474445, −8.957398349723348297134739665787, −8.478287867491043460030038424428, −7.33927356908189366838490077343, −6.41026247929694589950661301747, −4.81162938268634913473921831495, −3.93911997160820473683707040781, −2.98572404861385715991670418901, −2.05347084757411291618423909921, 1.06559652123727098244201505184, 2.79794225313774067029137272078, 4.14584942411494333995207454940, 4.79802228635547893216331992136, 5.99312531683457480015699663656, 7.03816142807988026512299499909, 8.055842192656439783950212172647, 9.066159720185176860497601292033, 9.595204439592862508166623476297, 10.69000337665265712285985069629

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