Properties

Label 2-21e2-9.4-c1-0-32
Degree $2$
Conductor $441$
Sign $0.5 + 0.866i$
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 + (0.592 − 1.62i)3-s + (0.613 − 1.06i)4-s + (0.673 − 1.16i)5-s + (1.50 − 0.264i)6-s + 2.83·8-s + (−2.29 − 1.92i)9-s + 1.18·10-s + (−0.826 − 1.43i)11-s + (−1.36 − 1.62i)12-s + (−1.68 + 2.91i)13-s + (−1.5 − 1.78i)15-s + (0.0209 + 0.0362i)16-s − 0.467·17-s + (0.458 − 2.59i)18-s + 3.22·19-s + ⋯
L(s)  = 1  + (0.310 + 0.538i)2-s + (0.342 − 0.939i)3-s + (0.306 − 0.531i)4-s + (0.301 − 0.521i)5-s + (0.612 − 0.107i)6-s + 1.00·8-s + (−0.766 − 0.642i)9-s + 0.374·10-s + (−0.249 − 0.431i)11-s + (−0.394 − 0.469i)12-s + (−0.467 + 0.809i)13-s + (−0.387 − 0.461i)15-s + (0.00523 + 0.00906i)16-s − 0.113·17-s + (0.107 − 0.612i)18-s + 0.740·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75766 - 1.01478i\)
\(L(\frac12)\) \(\approx\) \(1.75766 - 1.01478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.439 - 0.761i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.673 + 1.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.826 + 1.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.68 - 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 + (4.47 - 7.74i)T + (-11.5 - 19.9i)T^{2} \)
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 \( 1 - 9.23T + 37T^{2} \)
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 \( 1 + 0.573T + 53T^{2} \)
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 \( 1 + 2.04T + 73T^{2} \)
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 \( 1 + 9.08T + 89T^{2} \)
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

−11.31185257848052783515927405108, −9.790939495778211645390508060313, −9.214368916876326436177218124052, −7.80310984625730523904742488893, −7.39206281579855011543294069398, −6.08326337723307081809231840340, −5.66869753929676337997398512140, −4.26779620305591964594936952992, −2.48036989865594839413021645218, −1.25276497863972632445077942768, 2.39409908794547250465066989338, 3.09027669157189502421267644268, 4.27156308938759156736074494962, 5.20956273626953559492910596511, 6.63756365350569371823518561563, 7.75055692631221141879414158576, 8.545489981828227928352176189639, 9.854613043510095830955265467381, 10.42711645245486462281823922585, 11.05903817618152992095877060906

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