Properties

Label 2-21e2-441.142-c1-0-44
Degree $2$
Conductor $441$
Sign $0.392 + 0.919i$
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  + (0.454 − 1.15i)2-s + (1.71 − 0.239i)3-s + (0.333 + 0.309i)4-s + (−0.554 + 0.266i)5-s + (0.501 − 2.09i)6-s + (−1.17 − 2.37i)7-s + (2.74 − 1.32i)8-s + (2.88 − 0.821i)9-s + (0.0571 + 0.762i)10-s + (0.391 + 0.491i)11-s + (0.646 + 0.451i)12-s + (−0.676 + 1.72i)13-s + (−3.27 + 0.282i)14-s + (−0.886 + 0.590i)15-s + (−0.215 − 2.87i)16-s + (−0.396 + 0.368i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.818i)2-s + (0.990 − 0.138i)3-s + (0.166 + 0.154i)4-s + (−0.247 + 0.119i)5-s + (0.204 − 0.854i)6-s + (−0.444 − 0.895i)7-s + (0.972 − 0.468i)8-s + (0.961 − 0.273i)9-s + (0.0180 + 0.241i)10-s + (0.118 + 0.148i)11-s + (0.186 + 0.130i)12-s + (−0.187 + 0.477i)13-s + (−0.875 + 0.0756i)14-s + (−0.228 + 0.152i)15-s + (−0.0538 − 0.718i)16-s + (−0.0962 + 0.0893i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.392 + 0.919i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98644 - 1.31222i\)
\(L(\frac12)\) \(\approx\) \(1.98644 - 1.31222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.239i)T \)
7 \( 1 + (1.17 + 2.37i)T \)
good2 \( 1 + (-0.454 + 1.15i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (0.554 - 0.266i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.391 - 0.491i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.676 - 1.72i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (0.396 - 0.368i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.957 - 1.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.39 + 6.10i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-3.40 - 1.05i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.58 + 1.41i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (0.614 + 0.418i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (2.00 - 1.36i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (0.325 - 0.828i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (3.90 - 1.20i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.09 + 0.748i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (7.80 - 7.24i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (3.06 - 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.464 + 2.03i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.428 + 0.0646i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-3.84 - 6.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.35 + 11.0i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-4.94 - 12.5i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.579 + 1.00i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82377474958107642702553520163, −10.26990856272974984095666652561, −9.289299652088083345238868100693, −8.184675597646070183063092439401, −7.23855176141920402888870664760, −6.67147948355204448648581616092, −4.54379434199142941892459700756, −3.76480023758058948145972248324, −2.87703878681901034652603678373, −1.57138725562812197495629923124, 2.03064077083657076708960840501, 3.27913132731233968391792683713, 4.63317598333049860248753576349, 5.65955431038613485718617301210, 6.66826748965231471389265722847, 7.64288896287303285281526939495, 8.380627423919484751769870233200, 9.345847780711473612685253793521, 10.17380985909040033973165566135, 11.29579065093356730995005418751

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