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Properties

Label 2-21e2-147.101-c1-0-8
Degree $2$
Conductor $441$
Sign $0.373 - 0.927i$
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  + (2.07 + 0.815i)2-s + (2.18 + 2.03i)4-s + (2.05 + 1.40i)5-s + (−1.91 + 1.82i)7-s + (0.956 + 1.98i)8-s + (3.13 + 4.59i)10-s + (−0.173 − 1.15i)11-s + (1.20 − 0.960i)13-s + (−5.46 + 2.23i)14-s + (−0.0784 − 1.04i)16-s + (−2.49 − 0.769i)17-s + (0.129 − 0.0748i)19-s + (1.65 + 7.26i)20-s + (0.578 − 2.53i)22-s + (−0.369 − 1.19i)23-s + ⋯
L(s)  = 1  + (1.46 + 0.576i)2-s + (1.09 + 1.01i)4-s + (0.921 + 0.627i)5-s + (−0.722 + 0.690i)7-s + (0.338 + 0.702i)8-s + (0.991 + 1.45i)10-s + (−0.0523 − 0.347i)11-s + (0.333 − 0.266i)13-s + (−1.46 + 0.598i)14-s + (−0.0196 − 0.261i)16-s + (−0.604 − 0.186i)17-s + (0.0297 − 0.0171i)19-s + (0.370 + 1.62i)20-s + (0.123 − 0.540i)22-s + (−0.0769 − 0.249i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.373 - 0.927i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.373 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61866 + 1.76810i\)
\(L(\frac12)\) \(\approx\) \(2.61866 + 1.76810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 - 1.82i)T \)
good2 \( 1 + (-2.07 - 0.815i)T + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (-2.05 - 1.40i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.173 + 1.15i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-1.20 + 0.960i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (2.49 + 0.769i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.129 + 0.0748i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.369 + 1.19i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (0.805 - 0.183i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-8.87 - 5.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.79 + 8.16i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (6.87 - 3.31i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.17 + 4.41i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.43 - 3.65i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-2.28 + 2.46i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (8.87 - 6.04i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-2.83 - 3.05i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (4.19 - 7.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (14.2 + 3.26i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-10.1 + 3.98i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (0.550 + 0.953i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.71 + 4.65i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (8.96 + 1.35i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 9.69iT - 97T^{2} \)
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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

−11.61111668802000847857288404449, −10.45181395031296168747892453554, −9.577858785685762926273739799220, −8.464733182961906970753159436539, −7.03904071719054474683500086114, −6.26381003646639855147756441066, −5.80494043030784567598972888344, −4.67968853355833078583315106264, −3.30600419212663928407916234847, −2.50021412444815972362526590420, 1.62676221798387522366579541063, 2.94828418773563981531208718231, 4.15117893069995913839974535468, 4.93665567262650637767904401380, 6.06178166874958605232751506242, 6.67058625944003034193108445581, 8.247395060647241760970601031553, 9.529479486056471387159813129035, 10.14069857950797868779625699479, 11.20804472171161566824653570272

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