Properties

Label 2-21e2-147.131-c1-0-14
Degree $2$
Conductor $441$
Sign $-0.964 + 0.262i$
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  + (0.568 − 0.223i)2-s + (−1.19 + 1.10i)4-s + (−2.30 + 1.57i)5-s + (0.0953 − 2.64i)7-s + (−0.961 + 1.99i)8-s + (−0.962 + 1.41i)10-s + (0.355 − 2.35i)11-s + (−2.96 − 2.36i)13-s + (−0.536 − 1.52i)14-s + (0.141 − 1.89i)16-s + (−2.98 + 0.920i)17-s + (−5.52 − 3.19i)19-s + (1.01 − 4.43i)20-s + (−0.324 − 1.41i)22-s + (−2.41 + 7.82i)23-s + ⋯
L(s)  = 1  + (0.402 − 0.157i)2-s + (−0.596 + 0.553i)4-s + (−1.03 + 0.704i)5-s + (0.0360 − 0.999i)7-s + (−0.339 + 0.705i)8-s + (−0.304 + 0.446i)10-s + (0.107 − 0.710i)11-s + (−0.823 − 0.656i)13-s + (−0.143 − 0.407i)14-s + (0.0354 − 0.473i)16-s + (−0.724 + 0.223i)17-s + (−1.26 − 0.732i)19-s + (0.226 − 0.991i)20-s + (−0.0690 − 0.302i)22-s + (−0.503 + 1.63i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.964 + 0.262i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00792129 - 0.0593060i\)
\(L(\frac12)\) \(\approx\) \(0.00792129 - 0.0593060i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0953 + 2.64i)T \)
good2 \( 1 + (-0.568 + 0.223i)T + (1.46 - 1.36i)T^{2} \)
5 \( 1 + (2.30 - 1.57i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-0.355 + 2.35i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (2.96 + 2.36i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (2.98 - 0.920i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (5.52 + 3.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.41 - 7.82i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (8.18 + 1.86i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (-3.00 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.29 - 4.91i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-2.49 - 1.20i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-8.98 + 4.32i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.52 - 3.87i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (-0.503 - 0.542i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-2.77 - 1.89i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (8.06 - 8.69i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (1.63 + 2.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.32 + 0.986i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.43 + 3.31i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + (5.27 - 9.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.22 + 7.81i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (11.8 - 1.78i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 3.74iT - 97T^{2} \)
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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

−11.14866297213553197884002062209, −9.903741441398848559227198344602, −8.770720060063786289488561947610, −7.72452740198694681413595455081, −7.33785859750047108581801275886, −5.89753686369829190660332821363, −4.43187978333915075096455411744, −3.85383485137412120834060576284, −2.80594500032750603173821664824, −0.03200717281997268029460932388, 2.18202446853057146162310991635, 4.23446286729220273411691576354, 4.51084548037325553814084355476, 5.72870680579247750142783959298, 6.75161566280556186196134601424, 8.031836908513093163104679378941, 8.892087406854099123873228561412, 9.491200853903338087499482845219, 10.66450687262197975940900252529, 11.79125436021007331151010582346

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