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Properties

Label 2-21e2-147.101-c1-0-10
Degree $2$
Conductor $441$
Sign $0.998 + 0.0614i$
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  + (1.02 + 0.401i)2-s + (−0.580 − 0.538i)4-s + (1.00 + 0.687i)5-s + (2.62 − 0.364i)7-s + (−1.33 − 2.76i)8-s + (0.755 + 1.10i)10-s + (0.0890 + 0.590i)11-s + (2.95 − 2.35i)13-s + (2.82 + 0.679i)14-s + (−0.133 − 1.78i)16-s + (2.61 + 0.808i)17-s + (0.127 − 0.0734i)19-s + (−0.214 − 0.941i)20-s + (−0.146 + 0.640i)22-s + (2.07 + 6.71i)23-s + ⋯
L(s)  = 1  + (0.723 + 0.283i)2-s + (−0.290 − 0.269i)4-s + (0.450 + 0.307i)5-s + (0.990 − 0.137i)7-s + (−0.470 − 0.977i)8-s + (0.238 + 0.350i)10-s + (0.0268 + 0.178i)11-s + (0.819 − 0.653i)13-s + (0.755 + 0.181i)14-s + (−0.0334 − 0.446i)16-s + (0.635 + 0.195i)17-s + (0.0292 − 0.0168i)19-s + (−0.0480 − 0.210i)20-s + (−0.0311 + 0.136i)22-s + (0.431 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.998 + 0.0614i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.998 + 0.0614i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12929 - 0.0654450i\)
\(L(\frac12)\) \(\approx\) \(2.12929 - 0.0654450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.364i)T \)
good2 \( 1 + (-1.02 - 0.401i)T + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (-1.00 - 0.687i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.0890 - 0.590i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-2.95 + 2.35i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-2.61 - 0.808i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.127 + 0.0734i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.07 - 6.71i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-2.75 + 0.628i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (4.54 + 2.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.323 - 0.300i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (4.98 - 2.39i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-1.07 - 0.517i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.29 + 3.30i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (8.63 - 9.30i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (1.25 - 0.858i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (9.59 + 10.3i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (1.69 - 2.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.84 + 1.56i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (14.5 - 5.69i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-5.62 - 9.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.09 - 5.14i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (3.27 + 0.493i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−11.07926314507972821180382074210, −10.26109339346921102931162649797, −9.403452246441071383468549934053, −8.302313215447557977644546195603, −7.31626400082150071699502593648, −6.07730122234566828517903319570, −5.45332617919027766933063688560, −4.42103857433293739910532267498, −3.28213319612047592984934893276, −1.42012989970481559285994747240, 1.69835457465879813039446547194, 3.17175067727806713209055152401, 4.39384720924330396785213641291, 5.15853817449908670779532976428, 6.11360234420267403905945012349, 7.53973749545705054587993514855, 8.632281898566481898569022304130, 9.034001956269373956723837921271, 10.46679398187604612527311370522, 11.35120484489233600227611763501

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