Properties

Label 2-21e2-441.142-c1-0-46
Degree $2$
Conductor $441$
Sign $-0.749 + 0.661i$
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  + (0.485 − 1.23i)2-s + (0.455 − 1.67i)3-s + (0.173 + 0.161i)4-s + (−2.77 + 1.33i)5-s + (−1.84 − 1.37i)6-s + (2.57 − 0.627i)7-s + (2.67 − 1.28i)8-s + (−2.58 − 1.52i)9-s + (0.305 + 4.07i)10-s + (−3.53 − 4.43i)11-s + (0.348 − 0.216i)12-s + (1.73 − 4.41i)13-s + (0.470 − 3.48i)14-s + (0.969 + 5.24i)15-s + (−0.259 − 3.46i)16-s + (−2.81 + 2.61i)17-s + ⋯
L(s)  = 1  + (0.343 − 0.873i)2-s + (0.262 − 0.964i)3-s + (0.0868 + 0.0805i)4-s + (−1.24 + 0.597i)5-s + (−0.753 − 0.560i)6-s + (0.971 − 0.237i)7-s + (0.946 − 0.455i)8-s + (−0.861 − 0.507i)9-s + (0.0966 + 1.29i)10-s + (−1.06 − 1.33i)11-s + (0.100 − 0.0626i)12-s + (0.480 − 1.22i)13-s + (0.125 − 0.930i)14-s + (0.250 + 1.35i)15-s + (−0.0648 − 0.865i)16-s + (−0.683 + 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585170 - 1.54671i\)
\(L(\frac12)\) \(\approx\) \(0.585170 - 1.54671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.455 + 1.67i)T \)
7 \( 1 + (-2.57 + 0.627i)T \)
good2 \( 1 + (-0.485 + 1.23i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (2.77 - 1.33i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (3.53 + 4.43i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.73 + 4.41i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (2.81 - 2.61i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.03 + 1.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.366 + 1.60i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-9.59 - 2.96i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (1.22 - 2.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.45 - 1.68i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-5.89 - 4.01i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (3.27 - 2.23i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (3.63 - 9.26i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-4.16 + 1.28i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (7.34 - 5.01i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (2.44 - 2.26i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (0.887 - 1.53i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.76 + 12.1i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.964 - 0.145i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (0.278 + 0.481i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.13 + 5.43i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (0.772 + 1.96i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (1.28 - 2.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.93058030801041827552800940816, −10.64458616441261706505715263249, −8.430177249948105864385234710538, −8.046195818454747485616925483647, −7.36748731391095127780198877831, −6.16343177985771188587825302761, −4.64175801681432163597924320884, −3.28388590069038584985608907483, −2.76590104227003905449615700601, −0.959926201497298914211892773063, 2.20971091539367531898964051216, 4.28323271220343357639754864361, 4.58686928378326254905407426104, 5.46001062756365228114639451777, 6.98714739341242435299747252135, 7.892359631374058864099561481289, 8.422509653907481764486181870438, 9.561510463261339284283473157787, 10.68093322402984344826801546167, 11.44189368570996614261845585568

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