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Properties

Label 2-21e2-49.36-c1-0-10
Degree $2$
Conductor $441$
Sign $0.00272 - 0.999i$
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  + (0.243 + 1.06i)2-s + (0.718 − 0.346i)4-s + (2.19 + 2.74i)5-s + (0.626 + 2.57i)7-s + (1.91 + 2.39i)8-s + (−2.40 + 3.01i)10-s + (−0.448 − 1.96i)11-s + (−1.56 − 6.86i)13-s + (−2.59 + 1.29i)14-s + (−1.10 + 1.38i)16-s + (−3.40 − 1.63i)17-s + 0.124·19-s + (2.52 + 1.21i)20-s + (1.99 − 0.959i)22-s + (−4.63 + 2.23i)23-s + ⋯
L(s)  = 1  + (0.172 + 0.755i)2-s + (0.359 − 0.173i)4-s + (0.979 + 1.22i)5-s + (0.236 + 0.971i)7-s + (0.676 + 0.847i)8-s + (−0.759 + 0.952i)10-s + (−0.135 − 0.592i)11-s + (−0.434 − 1.90i)13-s + (−0.693 + 0.346i)14-s + (−0.275 + 0.345i)16-s + (−0.825 − 0.397i)17-s + 0.0286·19-s + (0.564 + 0.271i)20-s + (0.424 − 0.204i)22-s + (−0.965 + 0.465i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00272 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.00272 - 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.00272 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43894 + 1.43503i\)
\(L(\frac12)\) \(\approx\) \(1.43894 + 1.43503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.626 - 2.57i)T \)
good2 \( 1 + (-0.243 - 1.06i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (-2.19 - 2.74i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (0.448 + 1.96i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (1.56 + 6.86i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (3.40 + 1.63i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 0.124T + 19T^{2} \)
23 \( 1 + (4.63 - 2.23i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (0.781 + 0.376i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 6.42T + 31T^{2} \)
37 \( 1 + (-4.76 - 2.29i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (4.77 + 5.98i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-2.89 + 3.63i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.715 - 3.13i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.49 + 0.717i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-4.33 + 5.43i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (10.5 + 5.08i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + 5.05T + 67T^{2} \)
71 \( 1 + (-3.71 + 1.79i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.41 - 6.18i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + (-2.35 + 10.3i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.60 - 7.04i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−11.13160653966641247328444683747, −10.50001348830992000311660287616, −9.696765129438281160268885853586, −8.348024368788060890357667432528, −7.55723185594008132318703113255, −6.43589045411631726183774069159, −5.85328903588530477900849596359, −5.15260003895514027406299806194, −2.96645550231811144327056255564, −2.22874984553945184462480053045, 1.44197211499489451978038367723, 2.25815486848187856543291804103, 4.27749896778232214641953210799, 4.56149729666944834917460718630, 6.26515659363927230041372859600, 7.09221676792265551389036668189, 8.276989806567005957957229953309, 9.430631686374838207954225540526, 9.975308218736927146433542145608, 10.94606660807508249496476261930

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