↑

Properties

Label 2-21e2-63.16-c1-0-29
Degree $2$
Conductor $441$
Sign $0.873 + 0.487i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.119 + 0.207i)2-s + (1.71 − 0.272i)3-s + (0.971 − 1.68i)4-s + 1.18·5-s + (0.260 + 0.321i)6-s + 0.942·8-s + (2.85 − 0.931i)9-s + (0.141 + 0.244i)10-s − 3.70·11-s + (1.20 − 3.14i)12-s + (−0.5 − 0.866i)13-s + (2.02 − 0.321i)15-s + (−1.83 − 3.16i)16-s + (3.47 + 6.01i)17-s + (0.533 + 0.479i)18-s + (−0.971 + 1.68i)19-s + ⋯
L(s)  = 1  + (0.0845 + 0.146i)2-s + (0.987 − 0.157i)3-s + (0.485 − 0.841i)4-s + 0.528·5-s + (0.106 + 0.131i)6-s + 0.333·8-s + (0.950 − 0.310i)9-s + (0.0446 + 0.0774i)10-s − 1.11·11-s + (0.347 − 0.907i)12-s + (−0.138 − 0.240i)13-s + (0.522 − 0.0830i)15-s + (−0.457 − 0.792i)16-s + (0.841 + 1.45i)17-s + (0.125 + 0.112i)18-s + (−0.222 + 0.385i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.873 + 0.487i$
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: \((2,\ 441,\ (\ :1/2),\ 0.873 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23545 - 0.581892i\)
\(L(\frac12)\) \(\approx\) \(2.23545 - 0.581892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.272i)T \)
7 \( 1 \)
good2 \( 1 + (-0.119 - 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.18T + 5T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.47 - 6.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.971 - 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.60T + 23T^{2} \)
29 \( 1 + (0.119 - 0.207i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.830 - 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.09 - 8.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.11 - 1.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.80 - 10.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.30 - 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.80 - 6.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 + (7.57 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 + 6.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.37 - 2.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61223019526996754047899565839, −10.19817157704785087693120749228, −9.406072908948473834257274910423, −8.103405766153312586192027076795, −7.57045855627139334355605988479, −6.21377668916765105320636046516, −5.56962810906274523851419656299, −4.10073594017250321008704666569, −2.62301239629133119775779063264, −1.62998602028642733219654517237, 2.15880911901928764762469436026, 2.90252439774174758763314600069, 4.09406272915861283726589052920, 5.33608891328725984526118772991, 6.81834782256230203733435564747, 7.71662633276882773905175972623, 8.274381043234263801983211197546, 9.511823867792781822509567723673, 10.09004293007223472001561322608, 11.21951843364043184747286680776

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