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Properties

Label 2-21e2-21.20-c3-0-27
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  + 0.748i·2-s + 7.43·4-s + 10.8·5-s + 11.5i·8-s + 8.13i·10-s − 51.8i·11-s − 32.1i·13-s + 50.8·16-s + 81.4·17-s + 0.0485i·19-s + 80.7·20-s + 38.8·22-s − 89.2i·23-s − 7.16·25-s + 24.1·26-s + ⋯
L(s)  = 1  + 0.264i·2-s + 0.929·4-s + 0.970·5-s + 0.511i·8-s + 0.257i·10-s − 1.42i·11-s − 0.686i·13-s + 0.794·16-s + 1.16·17-s + 0.000586i·19-s + 0.902·20-s + 0.376·22-s − 0.809i·23-s − 0.0572·25-s + 0.181·26-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

\(L(2)\) \(\approx\) \(3.020531106\)
\(L(\frac12)\) \(\approx\) \(3.020531106\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.748iT - 8T^{2} \)
5 \( 1 - 10.8T + 125T^{2} \)
11 \( 1 + 51.8iT - 1.33e3T^{2} \)
13 \( 1 + 32.1iT - 2.19e3T^{2} \)
17 \( 1 - 81.4T + 4.91e3T^{2} \)
19 \( 1 - 0.0485iT - 6.85e3T^{2} \)
23 \( 1 + 89.2iT - 1.21e4T^{2} \)
29 \( 1 + 175. iT - 2.43e4T^{2} \)
31 \( 1 - 215. iT - 2.97e4T^{2} \)
37 \( 1 - 64.5T + 5.06e4T^{2} \)
41 \( 1 + 411.T + 6.89e4T^{2} \)
43 \( 1 + 234.T + 7.95e4T^{2} \)
47 \( 1 - 632.T + 1.03e5T^{2} \)
53 \( 1 + 265. iT - 1.48e5T^{2} \)
59 \( 1 - 351.T + 2.05e5T^{2} \)
61 \( 1 - 778. iT - 2.26e5T^{2} \)
67 \( 1 + 196.T + 3.00e5T^{2} \)
71 \( 1 - 142. iT - 3.57e5T^{2} \)
73 \( 1 - 780. iT - 3.89e5T^{2} \)
79 \( 1 - 1.28e3T + 4.93e5T^{2} \)
83 \( 1 - 235.T + 5.71e5T^{2} \)
89 \( 1 + 670.T + 7.04e5T^{2} \)
97 \( 1 - 655. iT - 9.12e5T^{2} \)
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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

−10.50111807200530857330116419053, −10.05580688459286736287321403919, −8.679799954411843181314188860464, −7.951785806413539789406984631278, −6.78582008971462343752649251545, −5.88826699371342901423450742720, −5.38795214908183952313007967346, −3.43562033796311900173312378013, −2.44605913056545081633274440601, −1.03700922907362904621671928705, 1.50399390311150455003408642091, 2.23812752142122884307037754602, 3.61351536217794674647919818685, 5.08909793914144832362270465596, 6.08402155143305887655457977358, 7.01080150064487969525590108289, 7.75900157725666868485667894750, 9.315840825090139994845979772011, 9.891030161176818162027386260019, 10.60600960819911199574221792430

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