Properties

Label 2-21e2-21.20-c3-0-32
Degree $2$
Conductor $441$
Sign $0.970 - 0.239i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.86i·2-s − 15.6·4-s + 12.7·5-s − 37.3i·8-s + 62.1i·10-s − 54.1i·11-s + 8.85i·13-s + 56.2·16-s − 68.9·17-s − 163. i·19-s − 200.·20-s + 263.·22-s − 93.9i·23-s + 37.9·25-s − 43.0·26-s + ⋯
L(s)  = 1  + 1.72i·2-s − 1.95·4-s + 1.14·5-s − 1.64i·8-s + 1.96i·10-s − 1.48i·11-s + 0.188i·13-s + 0.878·16-s − 0.983·17-s − 1.97i·19-s − 2.23·20-s + 2.55·22-s − 0.851i·23-s + 0.303·25-s − 0.324·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.410882887\)
\(L(\frac12)\) \(\approx\) \(1.410882887\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 - 4.86iT - 8T^{2} \)
5 \( 1 - 12.7T + 125T^{2} \)
11 \( 1 + 54.1iT - 1.33e3T^{2} \)
13 \( 1 - 8.85iT - 2.19e3T^{2} \)
17 \( 1 + 68.9T + 4.91e3T^{2} \)
19 \( 1 + 163. iT - 6.85e3T^{2} \)
23 \( 1 + 93.9iT - 1.21e4T^{2} \)
29 \( 1 + 119. iT - 2.43e4T^{2} \)
31 \( 1 + 98.8iT - 2.97e4T^{2} \)
37 \( 1 - 94.1T + 5.06e4T^{2} \)
41 \( 1 + 259.T + 6.89e4T^{2} \)
43 \( 1 - 5.01T + 7.95e4T^{2} \)
47 \( 1 - 57.3T + 1.03e5T^{2} \)
53 \( 1 - 470. iT - 1.48e5T^{2} \)
59 \( 1 + 225.T + 2.05e5T^{2} \)
61 \( 1 - 427. iT - 2.26e5T^{2} \)
67 \( 1 - 163.T + 3.00e5T^{2} \)
71 \( 1 + 79.8iT - 3.57e5T^{2} \)
73 \( 1 + 769. iT - 3.89e5T^{2} \)
79 \( 1 - 534.T + 4.93e5T^{2} \)
83 \( 1 + 438.T + 5.71e5T^{2} \)
89 \( 1 - 25.6T + 7.04e5T^{2} \)
97 \( 1 - 1.38e3iT - 9.12e5T^{2} \)
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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

−10.57226748267777185777815288385, −9.232629546096772291924012418964, −8.935365689241395186151420484293, −7.918512436960003789511852408686, −6.69150209524961459786784402938, −6.23387255209409937449683712671, −5.35049383616386508559796162554, −4.37410610998421755377302318891, −2.56048617392880886706100865533, −0.44598510318722759026548179506, 1.57907564301414753598083149391, 2.06996749742623482464346176783, 3.44320946790626596917797226243, 4.57252935369952333596068487636, 5.59066212245572597537386551072, 6.91231467783596344504543890415, 8.339812268258759801740043721572, 9.468781243537649288597415035291, 9.904598294337053780824817140436, 10.52481435102930202137016601533

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