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Properties

Label 2-21e2-147.41-c1-0-8
Degree $2$
Conductor $441$
Sign $0.438 - 0.898i$
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  + (−1.42 + 1.13i)2-s + (0.295 − 1.29i)4-s + (0.0720 − 0.0346i)5-s + (2.51 + 0.832i)7-s + (−0.533 − 1.10i)8-s + (−0.0632 + 0.131i)10-s + (0.0489 − 0.0390i)11-s + (3.06 − 2.44i)13-s + (−4.52 + 1.66i)14-s + (4.40 + 2.12i)16-s + (0.399 + 1.75i)17-s − 5.46i·19-s + (−0.0235 − 0.103i)20-s + (−0.0254 + 0.111i)22-s + (4.14 + 0.945i)23-s + ⋯
L(s)  = 1  + (−1.00 + 0.804i)2-s + (0.147 − 0.646i)4-s + (0.0322 − 0.0155i)5-s + (0.949 + 0.314i)7-s + (−0.188 − 0.391i)8-s + (−0.0200 + 0.0415i)10-s + (0.0147 − 0.0117i)11-s + (0.850 − 0.678i)13-s + (−1.20 + 0.446i)14-s + (1.10 + 0.530i)16-s + (0.0969 + 0.424i)17-s − 1.25i·19-s + (−0.00527 − 0.0231i)20-s + (−0.00541 + 0.0237i)22-s + (0.864 + 0.197i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790698 + 0.493999i\)
\(L(\frac12)\) \(\approx\) \(0.790698 + 0.493999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 - 0.832i)T \)
good2 \( 1 + (1.42 - 1.13i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (-0.0720 + 0.0346i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.0489 + 0.0390i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.06 + 2.44i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.399 - 1.75i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.46iT - 19T^{2} \)
23 \( 1 + (-4.14 - 0.945i)T + (20.7 + 9.97i)T^{2} \)
29 \( 1 + (-8.80 + 2.01i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 - 6.38iT - 31T^{2} \)
37 \( 1 + (-0.385 - 1.68i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (6.08 - 2.92i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-1.79 - 0.866i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.99 - 3.76i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.0985 + 0.0224i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (-3.52 - 1.69i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (3.20 - 0.731i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + (-2.96 - 0.677i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.291 - 0.232i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (-10.5 + 13.2i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (3.24 - 4.07i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−11.03646963080206080890987311822, −10.27152546289021758295976667154, −9.096356408283973591410692502172, −8.541066024611439600580298037115, −7.79460818852162086588413580534, −6.84917489441503869172951540247, −5.85224649857062242704396919284, −4.73597450735937286161770968290, −3.15194143455546097045178205713, −1.20050373892607358618824489680, 1.11106903653055022356584236758, 2.27375720588524750778745824245, 3.83653589561936187006407835256, 5.11611513400130813507820598135, 6.35377287200038677918770762223, 7.70957607150060308646216154037, 8.419690454265728963215471238533, 9.191980021630575144236340285723, 10.24381245438483193783736589044, 10.76714906055945360891913811025

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