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Properties

Label 2-21e2-7.6-c6-0-92
Degree $2$
Conductor $441$
Sign $-0.755 + 0.654i$
Analytic cond. $101.453$
Root an. cond. $10.0724$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.1·2-s + 109.·4-s − 79.5i·5-s + 601.·8-s − 1.04e3i·10-s − 822.·11-s + 2.42e3i·13-s + 913.·16-s − 7.79e3i·17-s − 6.68e3i·19-s − 8.72e3i·20-s − 1.08e4·22-s − 1.88e4·23-s + 9.29e3·25-s + 3.20e4i·26-s + ⋯
L(s)  = 1  + 1.64·2-s + 1.71·4-s − 0.636i·5-s + 1.17·8-s − 1.04i·10-s − 0.617·11-s + 1.10i·13-s + 0.223·16-s − 1.58i·17-s − 0.974i·19-s − 1.09i·20-s − 1.01·22-s − 1.54·23-s + 0.594·25-s + 1.82i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(101.453\)
Root analytic conductor: \(10.0724\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.011117608\)
\(L(\frac12)\) \(\approx\) \(3.011117608\)
\(L(4)\) 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 \( 1 \)
good2 \( 1 - 13.1T + 64T^{2} \)
5 \( 1 + 79.5iT - 1.56e4T^{2} \)
11 \( 1 + 822.T + 1.77e6T^{2} \)
13 \( 1 - 2.42e3iT - 4.82e6T^{2} \)
17 \( 1 + 7.79e3iT - 2.41e7T^{2} \)
19 \( 1 + 6.68e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.88e4T + 1.48e8T^{2} \)
29 \( 1 + 1.38e4T + 5.94e8T^{2} \)
31 \( 1 + 2.78e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.96e4T + 2.56e9T^{2} \)
41 \( 1 + 5.91e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.18e4T + 6.32e9T^{2} \)
47 \( 1 + 5.01e3iT - 1.07e10T^{2} \)
53 \( 1 + 1.86e5T + 2.21e10T^{2} \)
59 \( 1 + 2.25e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.44e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.35e5T + 9.04e10T^{2} \)
71 \( 1 + 9.62e4T + 1.28e11T^{2} \)
73 \( 1 - 2.75e5iT - 1.51e11T^{2} \)
79 \( 1 + 6.81e5T + 2.43e11T^{2} \)
83 \( 1 - 1.28e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.72e5iT - 4.96e11T^{2} \)
97 \( 1 - 6.20e5iT - 8.32e11T^{2} \)
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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

−9.802940353575350927574203562452, −8.941620528337529224044423206141, −7.61950332307290709386559522742, −6.71372138968365890544834279904, −5.71074105738107382982623044834, −4.80695448402089441694282173315, −4.24651273516113631718654963547, −2.93508333677536815954268197293, −2.00175247726547330687769730926, −0.30450378387122134621053266001, 1.73991201722493053836979266337, 2.92134824106961500338089522162, 3.64128844788389571492449407881, 4.69176732320002552551297882094, 5.86220552624178609098416738519, 6.22003265864145076871781251135, 7.54176720969002443078364611992, 8.325434808747636929473391935241, 10.07354360786726758903357935740, 10.63114966046095365935064345510

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