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Properties

Label 2-21e2-49.8-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.963 + 0.269i$
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  + (1.65 + 2.08i)2-s + (−1.12 + 4.95i)4-s + (−2.59 + 1.24i)5-s + (−1.31 − 2.29i)7-s + (−7.37 + 3.55i)8-s + (−6.90 − 3.32i)10-s + (2.50 + 3.14i)11-s + (2.30 + 2.89i)13-s + (2.58 − 6.54i)14-s + (−10.4 − 5.04i)16-s + (−0.667 − 2.92i)17-s − 3.46·19-s + (−3.25 − 14.2i)20-s + (−2.38 + 10.4i)22-s + (−0.254 + 1.11i)23-s + ⋯
L(s)  = 1  + (1.17 + 1.47i)2-s + (−0.564 + 2.47i)4-s + (−1.16 + 0.558i)5-s + (−0.497 − 0.867i)7-s + (−2.60 + 1.25i)8-s + (−2.18 − 1.05i)10-s + (0.755 + 0.947i)11-s + (0.639 + 0.802i)13-s + (0.691 − 1.74i)14-s + (−2.61 − 1.26i)16-s + (−0.161 − 0.709i)17-s − 0.794·19-s + (−0.727 − 3.18i)20-s + (−0.507 + 2.22i)22-s + (−0.0530 + 0.232i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.963 + 0.269i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.963 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.234284 - 1.70851i\)
\(L(\frac12)\) \(\approx\) \(0.234284 - 1.70851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 2.29i)T \)
good2 \( 1 + (-1.65 - 2.08i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (2.59 - 1.24i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.50 - 3.14i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.30 - 2.89i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.667 + 2.92i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + (0.254 - 1.11i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.836 - 3.66i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 7.66T + 31T^{2} \)
37 \( 1 + (-1.88 - 8.26i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-1.93 + 0.930i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-6.76 - 3.25i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.562 + 0.705i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.409 + 1.79i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.66 + 1.76i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (0.415 + 1.82i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (-3.15 + 13.8i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.63 + 4.55i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + (-1.57 + 1.97i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (3.41 - 4.27i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 2.32T + 97T^{2} \)
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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

−11.89204126473374472511442850889, −11.06631551134349983305326175834, −9.580249363969881291168710427898, −8.402935717118977313369527043062, −7.51142970442543177020773939384, −6.81146095558472940102075498335, −6.37696320348663702244573095001, −4.61617989262170403783946968746, −4.13953968491220535474123842689, −3.20644946071301962533843197952, 0.77981391703876208366095002279, 2.57084642391661765882846825235, 3.70273207706063288124235690957, 4.27961117968069897662567434578, 5.64524006206987471787953174380, 6.28728520356294566472647335689, 8.287434406274953659408119337314, 8.925858428646462465497177021468, 10.09407742749087389611438752291, 11.09744239219268569575536926145

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