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Properties

Label 2-21e2-63.25-c1-0-12
Degree $2$
Conductor $441$
Sign $0.967 + 0.253i$
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  − 0.239·2-s + (−1.09 − 1.34i)3-s − 1.94·4-s + (−0.590 + 1.02i)5-s + (0.260 + 0.321i)6-s + 0.942·8-s + (−0.619 + 2.93i)9-s + (0.141 − 0.244i)10-s + (1.85 + 3.20i)11-s + (2.11 + 2.61i)12-s + (−0.5 − 0.866i)13-s + (2.02 − 0.321i)15-s + 3.66·16-s + (3.47 − 6.01i)17-s + (0.148 − 0.701i)18-s + (−0.971 − 1.68i)19-s + ⋯
L(s)  = 1  − 0.169·2-s + (−0.629 − 0.776i)3-s − 0.971·4-s + (−0.264 + 0.457i)5-s + (0.106 + 0.131i)6-s + 0.333·8-s + (−0.206 + 0.978i)9-s + (0.0446 − 0.0774i)10-s + (0.558 + 0.967i)11-s + (0.611 + 0.754i)12-s + (−0.138 − 0.240i)13-s + (0.522 − 0.0830i)15-s + 0.915·16-s + (0.841 − 1.45i)17-s + (0.0349 − 0.165i)18-s + (−0.222 − 0.385i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.967 + 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778533 - 0.100120i\)
\(L(\frac12)\) \(\approx\) \(0.778533 - 0.100120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.09 + 1.34i)T \)
7 \( 1 \)
good2 \( 1 + 0.239T + 2T^{2} \)
5 \( 1 + (0.590 - 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.85 - 3.20i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (-2.80 + 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.119 - 0.207i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
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 \( 1 - 5.82T + 47T^{2} \)
53 \( 1 + (-5.80 + 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.60T + 59T^{2} \)
61 \( 1 + 7.60T + 61T^{2} \)
67 \( 1 - 3.50T + 67T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 + (7.57 - 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.37T + 79T^{2} \)
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 \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19588906799989752327387152468, −10.13068542863295008524292564106, −9.375297502175271972132626008563, −8.206437095890359720485972128864, −7.35019497075926602000642090743, −6.60214917456059950221503665626, −5.23659024610299446090665615220, −4.50142330143137551517289606004, −2.83732570488504292474643333126, −0.948980920962569836788598051200, 0.891407145659966308056892614784, 3.63849217342501234715875153699, 4.21143269761757994171133612556, 5.43017540144501146058442154720, 6.08691504350663686541410982176, 7.71834117838168719293670627051, 8.769832325721090714390667979125, 9.221268573517451807790188355853, 10.30586044569088187745141121607, 10.94266160791660416265082022273

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