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Properties

Label 2-21e2-63.4-c1-0-17
Degree $2$
Conductor $441$
Sign $0.605 + 0.795i$
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  + (0.439 − 0.761i)2-s + (−1.11 − 1.32i)3-s + (0.613 + 1.06i)4-s + 1.34·5-s + (−1.5 + 0.264i)6-s + 2.83·8-s + (−0.520 + 2.95i)9-s + (0.592 − 1.02i)10-s + 1.65·11-s + (0.726 − 1.99i)12-s + (1.68 − 2.91i)13-s + (−1.5 − 1.78i)15-s + (0.0209 − 0.0362i)16-s + (−0.233 + 0.405i)17-s + (2.02 + 1.69i)18-s + (1.61 + 2.79i)19-s + ⋯
L(s)  = 1  + (0.310 − 0.538i)2-s + (−0.642 − 0.766i)3-s + (0.306 + 0.531i)4-s + 0.602·5-s + (−0.612 + 0.107i)6-s + 1.00·8-s + (−0.173 + 0.984i)9-s + (0.187 − 0.324i)10-s + 0.498·11-s + (0.209 − 0.576i)12-s + (0.467 − 0.809i)13-s + (−0.387 − 0.461i)15-s + (0.00523 − 0.00906i)16-s + (−0.0567 + 0.0982i)17-s + (0.476 + 0.399i)18-s + (0.370 + 0.641i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 441,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54687 - 0.766828i\)
\(L(\frac12)\) \(\approx\) \(1.54687 - 0.766828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 + 1.32i)T \)
7 \( 1 \)
good2 \( 1 + (-0.439 + 0.761i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.34T + 5T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.233 - 0.405i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + (3.13 + 5.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.61 + 7.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.70 - 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 3.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.67 - 8.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.286 + 0.497i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.19 - 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.81 - 6.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.298 + 0.516i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.554T + 71T^{2} \)
73 \( 1 + (1.02 - 1.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.52 - 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.54 + 7.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.949 - 1.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.20483320511531534068090446810, −10.47206367295542940216042914310, −9.281272626305013244012971528598, −7.980938120989918386908790374740, −7.31625391447585698542672386689, −6.20360106806608713046304893021, −5.38436388529616352901938425534, −3.95165543361171249301758955541, −2.61316087231832447365979469722, −1.40000977447550774505957249211, 1.50792248904301098063192232970, 3.50526435110638626345219114257, 4.92223315931463880684090212539, 5.38230812697238564514380319898, 6.61834785531629225917214233497, 6.94858058933185393775940421850, 8.853756626599044319479436465524, 9.462992366450406661789618286963, 10.48549682056247329527793168971, 11.09418933271149358609166866000

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