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Properties

Label 2-21e2-63.38-c1-0-3
Degree $2$
Conductor $441$
Sign $0.730 + 0.683i$
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  + 2.70i·2-s + (−0.572 + 1.63i)3-s − 5.33·4-s + (−0.601 + 1.04i)5-s + (−4.42 − 1.54i)6-s − 9.04i·8-s + (−2.34 − 1.87i)9-s + (−2.82 − 1.62i)10-s + (2.15 − 1.24i)11-s + (3.05 − 8.72i)12-s + (−1.63 + 0.942i)13-s + (−1.35 − 1.57i)15-s + 13.8·16-s + (−0.601 + 1.04i)17-s + (5.06 − 6.35i)18-s + (−6.46 + 3.73i)19-s + ⋯
L(s)  = 1  + 1.91i·2-s + (−0.330 + 0.943i)3-s − 2.66·4-s + (−0.268 + 0.465i)5-s + (−1.80 − 0.632i)6-s − 3.19i·8-s + (−0.781 − 0.623i)9-s + (−0.892 − 0.515i)10-s + (0.649 − 0.374i)11-s + (0.881 − 2.51i)12-s + (−0.452 + 0.261i)13-s + (−0.350 − 0.407i)15-s + 3.45·16-s + (−0.145 + 0.252i)17-s + (1.19 − 1.49i)18-s + (−1.48 + 0.856i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.730 + 0.683i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.730 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.390605 - 0.154176i\)
\(L(\frac12)\) \(\approx\) \(0.390605 - 0.154176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.572 - 1.63i)T \)
7 \( 1 \)
good2 \( 1 - 2.70iT - 2T^{2} \)
5 \( 1 + (0.601 - 1.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.15 + 1.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.63 - 0.942i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.46 - 3.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.63 + 1.52i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.173 + 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.50iT - 31T^{2} \)
37 \( 1 + (0.865 + 1.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.36 + 5.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00656 + 0.0113i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.43T + 47T^{2} \)
53 \( 1 + (8.58 + 4.95i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + 5.15T + 67T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (7.51 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.49T + 79T^{2} \)
83 \( 1 + (1.60 - 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.98 + 6.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.06 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.95472758171509016100035401769, −10.68864933459413299953968750161, −9.882180052455465189614507573717, −8.867296781112517330735336116805, −8.333794299002798779160987279465, −7.05905578445537375966390091413, −6.34124186221951019596267061786, −5.50928930057528075783543680792, −4.40352795367322602824754446541, −3.67315383226873542024966220782, 0.28125147284736339219214285208, 1.70139233572272570846475163050, 2.74261777960792390747220527488, 4.22885171393018562433919003676, 5.03935915879574281530321295651, 6.49029186383445531822791825349, 7.949828825245775153458941788251, 8.724894739716198619399765206010, 9.621974987482046447566996617803, 10.64228358653769181250788009227

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