Properties

Label 2-21e2-49.22-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.159 - 0.987i$
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  + (0.5 − 0.240i)2-s + (−1.05 + 1.32i)4-s + (−0.321 + 1.40i)5-s + (2.57 − 0.588i)7-s + (−0.455 + 1.99i)8-s + (0.178 + 0.781i)10-s + (−3.32 + 1.60i)11-s + (−5.25 + 2.52i)13-s + (1.14 − 0.915i)14-s + (−0.500 − 2.19i)16-s + (1.81 + 2.27i)17-s + 1.93·19-s + (−1.52 − 1.91i)20-s + (−1.27 + 1.60i)22-s + (−0.815 + 1.02i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.170i)2-s + (−0.527 + 0.661i)4-s + (−0.143 + 0.630i)5-s + (0.974 − 0.222i)7-s + (−0.161 + 0.706i)8-s + (0.0564 + 0.247i)10-s + (−1.00 + 0.482i)11-s + (−1.45 + 0.701i)13-s + (0.306 − 0.244i)14-s + (−0.125 − 0.547i)16-s + (0.440 + 0.552i)17-s + 0.444·19-s + (−0.340 − 0.427i)20-s + (−0.272 + 0.341i)22-s + (−0.170 + 0.213i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798851 + 0.938376i\)
\(L(\frac12)\) \(\approx\) \(0.798851 + 0.938376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.57 + 0.588i)T \)
good2 \( 1 + (-0.5 + 0.240i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (0.321 - 1.40i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (3.32 - 1.60i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (5.25 - 2.52i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-1.81 - 2.27i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 + (0.815 - 1.02i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-4.92 - 6.17i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 + (1.13 + 1.42i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (1.69 - 7.41i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.475 - 2.08i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-4.02 + 1.93i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-8.85 + 11.0i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.43 + 6.28i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (1.95 + 2.45i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 1.04T + 67T^{2} \)
71 \( 1 + (-3.79 + 4.75i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.23 - 0.593i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + (4.33 + 2.08i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (8.48 + 4.08i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 1.35T + 97T^{2} \)
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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

−11.48535975814640161937398486343, −10.57352951471469824652638914510, −9.667145688750456281378856491918, −8.485440067493797681829761670769, −7.64677223024131153715308361251, −7.03583972210185579560548769002, −5.21702598209138893820224473424, −4.69080974514985826374556831385, −3.38889224175851233801621439773, −2.21956291785853385926931319298, 0.70661006873585351337156361840, 2.60363442661833080602422101610, 4.37000963961824166690264547728, 5.18513358002557348896410734431, 5.63822204577503443275232497041, 7.29330737142840852910520420697, 8.143970579701107935848249941438, 9.023444733562731936347987831999, 10.05687956425522629131531581670, 10.70719816259838418324658048807

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