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Properties

Label 2-21e2-63.58-c1-0-4
Degree $2$
Conductor $441$
Sign $-0.888 - 0.458i$
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-s + 1.73i·3-s − 4-s + (−0.5 − 0.866i)5-s + 1.73i·6-s − 3·8-s − 2.99·9-s + (−0.5 − 0.866i)10-s + (−2.5 + 4.33i)11-s − 1.73i·12-s + (−2.5 + 4.33i)13-s + (1.49 − 0.866i)15-s − 16-s + (1.5 + 2.59i)17-s − 2.99·18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.999i·3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + 0.707i·6-s − 1.06·8-s − 0.999·9-s + (−0.158 − 0.273i)10-s + (−0.753 + 1.30i)11-s − 0.499i·12-s + (−0.693 + 1.20i)13-s + (0.387 − 0.223i)15-s − 0.250·16-s + (0.363 + 0.630i)17-s − 0.707·18-s + (0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212143 + 0.874815i\)
\(L(\frac12)\) \(\approx\) \(0.212143 + 0.874815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
7 \( 1 \)
good2 \( 1 - T + 2T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.77571165427523761745811034438, −10.40692253208948842615237675603, −9.759990293972023114358226004503, −8.947118938118963554793137628289, −8.050452790144962242927775413890, −6.63708244465516104498097115985, −5.33256860122687064871953288635, −4.62499470423962819147532538854, −4.04673028940956616442490297138, −2.58159974912724903515731153491, 0.44116309322243106104866047721, 2.76867154788269676611066478745, 3.49025161554665098994571218138, 5.34611626000252588588114924133, 5.64282061840593164526493357085, 7.01334971871965931888962796787, 7.942170815032265332943663847739, 8.647589038592275372985870014402, 9.891365905234033957090596478698, 11.01144369866925413890849561797

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