Properties

Degree $2$
Conductor $441$
Sign $0.482 - 0.875i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.02 + 1.77i)2-s + (−0.608 − 1.62i)3-s + (−1.10 − 1.92i)4-s + (−0.0731 − 0.126i)5-s + (3.50 + 0.582i)6-s + 0.446·8-s + (−2.25 + 1.97i)9-s + 0.300·10-s + (−0.832 + 1.44i)11-s + (−2.43 + 2.96i)12-s + (−0.0999 − 0.173i)13-s + (−0.160 + 0.195i)15-s + (1.75 − 3.04i)16-s + 6.27·17-s + (−1.19 − 6.04i)18-s + 6.91·19-s + ⋯
L(s)  = 1  + (−0.726 + 1.25i)2-s + (−0.351 − 0.936i)3-s + (−0.554 − 0.960i)4-s + (−0.0327 − 0.0566i)5-s + (1.43 + 0.237i)6-s + 0.157·8-s + (−0.752 + 0.658i)9-s + 0.0949·10-s + (−0.250 + 0.434i)11-s + (−0.704 + 0.856i)12-s + (−0.0277 − 0.0480i)13-s + (−0.0415 + 0.0505i)15-s + (0.439 − 0.761i)16-s + 1.52·17-s + (−0.280 − 1.42i)18-s + 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.482 - 0.875i$
Motivic weight: \(1\)
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.482 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688862 + 0.407003i\)
\(L(\frac12)\) \(\approx\) \(0.688862 + 0.407003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.608 + 1.62i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 - 1.77i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.0731 + 0.126i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.832 - 1.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0999 + 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.27T + 17T^{2} \)
19 \( 1 - 6.91T + 19T^{2} \)
23 \( 1 + (-3.09 - 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.46 - 4.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.00T + 37T^{2} \)
41 \( 1 + (1.15 + 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.940 - 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.905 - 1.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.34T + 53T^{2} \)
59 \( 1 + (2.28 + 3.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.339 - 0.587i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + (6.39 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.75 - 6.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.06T + 89T^{2} \)
97 \( 1 + (-3.98 + 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.37954001021299503050060344128, −10.05532144331304836776166380925, −9.269454905380162178093900722254, −8.132666141030394749651948010549, −7.52700470703393831173871553875, −6.95152271060748559876831788936, −5.75390490151706609407586908642, −5.22319058933087873184558963855, −3.05712346734765522272753855155, −1.07693927570911668138385012274, 0.910934178483642533455547964147, 2.88314655155537217798899518053, 3.59362019363070514481821934305, 5.06163444205335974603285851599, 6.01240965042256625710456665262, 7.61713562333591418693071891446, 8.721424163090601402395092525951, 9.502507000185946895433978786962, 10.12944159618314310362276862376, 10.87142537649875306873268551942

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