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Properties

Label 2-21e2-21.20-c1-0-5
Degree $2$
Conductor $441$
Sign $0.860 - 0.508i$
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  + 0.414i·2-s + 1.82·4-s + 2.93·5-s + 1.58i·8-s + 1.21i·10-s + 4.82i·11-s − 2.93i·13-s + 3·16-s − 7.07·17-s − 5.86i·19-s + 5.35·20-s − 1.99·22-s − 2i·23-s + 3.58·25-s + 1.21·26-s + ⋯
L(s)  = 1  + 0.292i·2-s + 0.914·4-s + 1.31·5-s + 0.560i·8-s + 0.383i·10-s + 1.45i·11-s − 0.812i·13-s + 0.750·16-s − 1.71·17-s − 1.34i·19-s + 1.19·20-s − 0.426·22-s − 0.417i·23-s + 0.717·25-s + 0.238·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.860 - 0.508i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.860 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95595 + 0.534847i\)
\(L(\frac12)\) \(\approx\) \(1.95595 + 0.534847i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 - 0.414iT - 2T^{2} \)
5 \( 1 - 2.93T + 5T^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 + 5.86iT - 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 0.828iT - 29T^{2} \)
31 \( 1 - 5.86iT - 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 - 5.86T + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 - 5.86T + 59T^{2} \)
61 \( 1 + 1.21iT - 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 - 0.828iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 - 1.65T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 7.07iT - 97T^{2} \)
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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

−10.96798398471856688161771092837, −10.38046968805263656967104027830, −9.460037519816967523724871735098, −8.523349353286868070949716133981, −7.10987753088453204996606162631, −6.71484269538740013911907216201, −5.60644107097189058033372792074, −4.66692034548636840498003141407, −2.67463167110657769592876196361, −1.90607441545801680421054824736, 1.61764564249852980721179987264, 2.59586479729940079319977276241, 3.96453924116737751355353249677, 5.72023626143805448293462675793, 6.16664078910544433489574451100, 7.14447339648659723823607067696, 8.493197555373189547438942973327, 9.351230968105701448367265067997, 10.29033016986047027858086220981, 11.03123987990486918242385734858

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