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Properties

Label 2-21e2-147.104-c1-0-14
Degree $2$
Conductor $441$
Sign $0.989 - 0.147i$
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  + (0.947 + 0.755i)2-s + (−0.118 − 0.518i)4-s + (3.15 + 1.52i)5-s + (1.53 − 2.15i)7-s + (1.33 − 2.76i)8-s + (1.84 + 3.82i)10-s + (−0.959 − 0.765i)11-s + (−4.81 − 3.84i)13-s + (3.08 − 0.883i)14-s + (2.39 − 1.15i)16-s + (−0.101 + 0.445i)17-s + 8.04i·19-s + (0.414 − 1.81i)20-s + (−0.330 − 1.44i)22-s + (−1.94 + 0.443i)23-s + ⋯
L(s)  = 1  + (0.669 + 0.534i)2-s + (−0.0591 − 0.259i)4-s + (1.41 + 0.679i)5-s + (0.579 − 0.814i)7-s + (0.470 − 0.977i)8-s + (0.582 + 1.20i)10-s + (−0.289 − 0.230i)11-s + (−1.33 − 1.06i)13-s + (0.823 − 0.236i)14-s + (0.597 − 0.287i)16-s + (−0.0246 + 0.107i)17-s + 1.84i·19-s + (0.0926 − 0.405i)20-s + (−0.0705 − 0.309i)22-s + (−0.405 + 0.0924i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.989 - 0.147i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37226 + 0.175820i\)
\(L(\frac12)\) \(\approx\) \(2.37226 + 0.175820i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 + 2.15i)T \)
good2 \( 1 + (-0.947 - 0.755i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-3.15 - 1.52i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (0.959 + 0.765i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.81 + 3.84i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.101 - 0.445i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 8.04iT - 19T^{2} \)
23 \( 1 + (1.94 - 0.443i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-6.85 - 1.56i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 6.87iT - 31T^{2} \)
37 \( 1 + (-0.226 + 0.994i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-5.35 - 2.57i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (8.06 - 3.88i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.922 - 1.15i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (7.14 - 1.63i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-6.83 + 3.29i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (6.28 + 1.43i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 1.41T + 67T^{2} \)
71 \( 1 + (14.0 - 3.19i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.23 - 0.982i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 6.59T + 79T^{2} \)
83 \( 1 + (-3.24 - 4.06i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-8.27 - 10.3i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−10.76670536653207425739239725382, −10.11957756218760366632647653600, −9.871884094109929366131330266244, −8.135120476151902385573713635183, −7.20872167140962084545756761872, −6.25646490869361571989711892453, −5.50205142352619327525164866695, −4.66258284039593084798600609682, −3.13572380418150596217536068699, −1.56216165859924007129657255100, 2.05778525611358083836189314747, 2.57124060802740261341072074975, 4.73089866805674626071080375052, 4.85733581064152628251577569715, 6.09538898538093573876877157623, 7.41301404096876747374322825824, 8.664207014755422645505094793775, 9.266774901895640367758611769140, 10.17996138590828738755365109004, 11.48600321105428064938197555771

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