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Properties

Label 2-21e2-441.5-c1-0-29
Degree $2$
Conductor $441$
Sign $0.403 - 0.914i$
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  + (−1.47 + 1.17i)2-s + (1.71 − 0.261i)3-s + (0.346 − 1.51i)4-s + (3.14 + 2.14i)5-s + (−2.21 + 2.39i)6-s + (0.361 − 2.62i)7-s + (−0.363 − 0.754i)8-s + (2.86 − 0.896i)9-s + (−7.15 + 0.536i)10-s + (0.626 + 4.15i)11-s + (0.195 − 2.68i)12-s + (0.0779 + 0.517i)13-s + (2.54 + 4.28i)14-s + (5.94 + 2.84i)15-s + (4.22 + 2.03i)16-s + (3.64 − 3.37i)17-s + ⋯
L(s)  = 1  + (−1.04 + 0.831i)2-s + (0.988 − 0.151i)3-s + (0.173 − 0.758i)4-s + (1.40 + 0.959i)5-s + (−0.904 + 0.979i)6-s + (0.136 − 0.990i)7-s + (−0.128 − 0.266i)8-s + (0.954 − 0.298i)9-s + (−2.26 + 0.169i)10-s + (0.188 + 1.25i)11-s + (0.0564 − 0.775i)12-s + (0.0216 + 0.143i)13-s + (0.681 + 1.14i)14-s + (1.53 + 0.735i)15-s + (1.05 + 0.508i)16-s + (0.883 − 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.403 - 0.914i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.403 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21284 + 0.790416i\)
\(L(\frac12)\) \(\approx\) \(1.21284 + 0.790416i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.261i)T \)
7 \( 1 + (-0.361 + 2.62i)T \)
good2 \( 1 + (1.47 - 1.17i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (-3.14 - 2.14i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.626 - 4.15i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-0.0779 - 0.517i)T + (-12.4 + 3.83i)T^{2} \)
17 \( 1 + (-3.64 + 3.37i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (5.37 + 3.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.64 + 5.34i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (0.339 + 0.366i)T + (-2.16 + 28.9i)T^{2} \)
31 \( 1 - 5.17iT - 31T^{2} \)
37 \( 1 + (8.49 + 2.61i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (0.671 - 8.96i)T + (-40.5 - 6.11i)T^{2} \)
43 \( 1 + (0.239 + 3.19i)T + (-42.5 + 6.40i)T^{2} \)
47 \( 1 + (-2.76 - 3.47i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-3.24 - 10.5i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (-5.76 - 2.77i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (3.88 - 0.887i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (15.0 + 3.44i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.116 + 0.771i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 - 5.63T + 79T^{2} \)
83 \( 1 + (10.8 + 1.63i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (1.78 + 4.55i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-11.5 + 6.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.42831669194715362522237711005, −10.25063128551547183743030863693, −9.397925813894654182396284496375, −8.647734274402861862651418792710, −7.33242707559075286881495476348, −7.08262572426128725261164034699, −6.23724925381326698308305202899, −4.46515799188416992661412903672, −2.93496977442169707798145622840, −1.63508236974275624850087411908, 1.50151345611205493422253557818, 2.19929455262016727796643248685, 3.53524392447946562201043717270, 5.38674950916855289795808817642, 5.99369063063126071743222119790, 8.047727234635142757469140969936, 8.665315503444324598548406705818, 9.045572486623074707107458735779, 9.966269753334762054723731456622, 10.46387407369294039530230973735

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