Properties

Label 2-21e2-49.39-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.997 - 0.0672i$
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  + (−2.32 − 0.350i)2-s + (3.37 + 1.04i)4-s + (−0.0416 − 0.555i)5-s + (−1.55 − 2.13i)7-s + (−3.24 − 1.56i)8-s + (−0.0979 + 1.30i)10-s + (−0.266 − 0.678i)11-s + (−1.59 + 2.00i)13-s + (2.86 + 5.52i)14-s + (1.16 + 0.792i)16-s + (1.02 + 0.954i)17-s + (−2.19 − 3.80i)19-s + (0.437 − 1.91i)20-s + (0.381 + 1.67i)22-s + (−3.73 + 3.46i)23-s + ⋯
L(s)  = 1  + (−1.64 − 0.247i)2-s + (1.68 + 0.520i)4-s + (−0.0186 − 0.248i)5-s + (−0.588 − 0.808i)7-s + (−1.14 − 0.552i)8-s + (−0.0309 + 0.413i)10-s + (−0.0802 − 0.204i)11-s + (−0.443 + 0.556i)13-s + (0.766 + 1.47i)14-s + (0.290 + 0.198i)16-s + (0.249 + 0.231i)17-s + (−0.503 − 0.872i)19-s + (0.0978 − 0.428i)20-s + (0.0812 + 0.356i)22-s + (−0.778 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.997 - 0.0672i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.997 - 0.0672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00504534 + 0.149766i\)
\(L(\frac12)\) \(\approx\) \(0.00504534 + 0.149766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 2.13i)T \)
good2 \( 1 + (2.32 + 0.350i)T + (1.91 + 0.589i)T^{2} \)
5 \( 1 + (0.0416 + 0.555i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (0.266 + 0.678i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (1.59 - 2.00i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-1.02 - 0.954i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (2.19 + 3.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.73 - 3.46i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.72 + 7.55i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.93 - 3.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.82 - 2.10i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (9.91 + 4.77i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (9.13 - 4.39i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (11.5 + 1.73i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (3.25 + 1.00i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.777 - 10.3i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-9.94 + 3.06i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.99 + 8.73i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (10.3 - 1.55i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (3.15 + 5.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.50 - 1.89i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-0.183 + 0.467i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 1.13T + 97T^{2} \)
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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

−10.26828435814671623216119536656, −9.894624070374125966436375607939, −8.887253222745856853623895635188, −8.167329105213649480298125984010, −7.13887064274220843089393328260, −6.49371941273051634394639203671, −4.76631816570825740060994700896, −3.26644167780391452976094433327, −1.74235217627772522257040281722, −0.15367823230026402832123461282, 1.88981890187416353948388094978, 3.20959049166009833557761002265, 5.18594850006614609711112374579, 6.41380605327621056120235685301, 7.08463931655606577830879005201, 8.273196028512873843965240676844, 8.705195919649256962067295672665, 9.981099288327131492758831511702, 10.11903337445467943479733429264, 11.27730956110568449390202456051

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