Properties

Label 2-21e2-63.16-c1-0-32
Degree $2$
Conductor $441$
Sign $-0.991 - 0.126i$
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  + (−0.673 − 1.16i)2-s + (−0.592 − 1.62i)3-s + (0.0923 − 0.160i)4-s + 2.53·5-s + (−1.5 + 1.78i)6-s − 2.94·8-s + (−2.29 + 1.92i)9-s + (−1.70 − 2.95i)10-s + 0.467·11-s + (−0.315 − 0.0555i)12-s + (−2.91 − 5.04i)13-s + (−1.50 − 4.12i)15-s + (1.79 + 3.11i)16-s + (−1.93 − 3.35i)17-s + (3.79 + 1.38i)18-s + (1.09 − 1.89i)19-s + ⋯
L(s)  = 1  + (−0.476 − 0.825i)2-s + (−0.342 − 0.939i)3-s + (0.0461 − 0.0800i)4-s + 1.13·5-s + (−0.612 + 0.729i)6-s − 1.04·8-s + (−0.766 + 0.642i)9-s + (−0.539 − 0.934i)10-s + 0.141·11-s + (−0.0909 − 0.0160i)12-s + (−0.807 − 1.39i)13-s + (−0.387 − 1.06i)15-s + (0.449 + 0.778i)16-s + (−0.470 − 0.814i)17-s + (0.895 + 0.325i)18-s + (0.250 − 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0611819 + 0.964099i\)
\(L(\frac12)\) \(\approx\) \(0.0611819 + 0.964099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.592 + 1.62i)T \)
7 \( 1 \)
good2 \( 1 + (0.673 + 1.16i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.53T + 5T^{2} \)
11 \( 1 - 0.467T + 11T^{2} \)
13 \( 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 + 3.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.106T + 23T^{2} \)
29 \( 1 + (4.39 - 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.84 + 6.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.66 - 4.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.358 - 0.620i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.368 - 0.637i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.81 + 8.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (-5.13 - 8.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.36 + 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.05 + 7.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.78336319667926171534473709804, −9.773536202202482338526098625395, −9.210113750748524708554255227749, −7.925274306566375665720411836748, −6.87489739737295517622900026910, −5.87021341142985628931316088727, −5.22217770478616139700750613429, −2.86044548557032563375394780558, −2.10457517115587742351704721888, −0.70848730091501298937955083459, 2.29588297420373919613456075033, 3.87056333820067728326078018899, 5.12023878697176692742483840359, 6.16955009059549070888603120061, 6.67698762694508223778481924925, 8.043330516556755910433343312319, 9.089892889928239528779200000732, 9.568849517265836653385594381550, 10.36990841932979565135731042448, 11.58212222698234384447234915950

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