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Properties

Label 2-21e2-63.4-c1-0-8
Degree $2$
Conductor $441$
Sign $0.983 - 0.183i$
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  + (−0.849 + 1.47i)2-s + (−1.64 − 0.545i)3-s + (−0.444 − 0.769i)4-s − 3.58·5-s + (2.19 − 1.95i)6-s − 1.88·8-s + (2.40 + 1.79i)9-s + (3.04 − 5.28i)10-s − 2.81·11-s + (0.310 + 1.50i)12-s + (−0.5 + 0.866i)13-s + (5.89 + 1.95i)15-s + (2.49 − 4.31i)16-s + (2.05 − 3.56i)17-s + (−4.68 + 2.01i)18-s + (0.444 + 0.769i)19-s + ⋯
L(s)  = 1  + (−0.600 + 1.04i)2-s + (−0.949 − 0.314i)3-s + (−0.222 − 0.384i)4-s − 1.60·5-s + (0.897 − 0.798i)6-s − 0.667·8-s + (0.801 + 0.597i)9-s + (0.964 − 1.67i)10-s − 0.847·11-s + (0.0897 + 0.435i)12-s + (−0.138 + 0.240i)13-s + (1.52 + 0.505i)15-s + (0.623 − 1.07i)16-s + (0.498 − 0.863i)17-s + (−1.10 + 0.475i)18-s + (0.101 + 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.983 - 0.183i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.983 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354583 + 0.0327878i\)
\(L(\frac12)\) \(\approx\) \(0.354583 + 0.0327878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 + 0.545i)T \)
7 \( 1 \)
good2 \( 1 + (0.849 - 1.47i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.58T + 5T^{2} \)
11 \( 1 + 2.81T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.05 + 3.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.444 - 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.87T + 23T^{2} \)
29 \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.49 - 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.38 + 4.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.70 + 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0618 + 0.107i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.15 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + (-5.32 + 9.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.54 + 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.05 - 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.80 + 8.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.66 + 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.20306162865362672410745342865, −10.35030366832410169286725605973, −9.001271331534355348646057216513, −8.084313479658693917882395140161, −7.28269488843603556697917739703, −6.97940733043976384084285845572, −5.57770760351181375658980786695, −4.70108058227496479164179504753, −3.17285578977244598190542924527, −0.44495275735223829949425951744, 0.887013104025779839170708036383, 2.97232722948196740474873912056, 4.03554219161213524718551211512, 5.16065866695518696909361213231, 6.41842342263829961144584954651, 7.64446204078661168171069134534, 8.441134526147027625770782853316, 9.677458486929656422802148768792, 10.43446109344799436383524391778, 11.19411396464162412101084154312

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