Properties

Label 2-21e2-49.43-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.0714 - 0.997i$
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  + (−1.64 + 2.05i)2-s + (−1.09 − 4.79i)4-s + (−3.33 − 1.60i)5-s + (−2.39 + 1.11i)7-s + (6.91 + 3.33i)8-s + (8.78 − 4.22i)10-s + (1.47 − 1.84i)11-s + (−0.990 + 1.24i)13-s + (1.63 − 6.76i)14-s + (−9.33 + 4.49i)16-s + (0.263 − 1.15i)17-s + 4.37·19-s + (−4.05 + 17.7i)20-s + (1.38 + 6.05i)22-s + (1.93 + 8.46i)23-s + ⋯
L(s)  = 1  + (−1.15 + 1.45i)2-s + (−0.547 − 2.39i)4-s + (−1.49 − 0.719i)5-s + (−0.906 + 0.422i)7-s + (2.44 + 1.17i)8-s + (2.77 − 1.33i)10-s + (0.443 − 0.556i)11-s + (−0.274 + 0.344i)13-s + (0.436 − 1.80i)14-s + (−2.33 + 1.12i)16-s + (0.0637 − 0.279i)17-s + 1.00·19-s + (−0.907 + 3.97i)20-s + (0.294 + 1.29i)22-s + (0.402 + 1.76i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0714 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0714 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0714 - 0.997i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0714 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290656 + 0.312212i\)
\(L(\frac12)\) \(\approx\) \(0.290656 + 0.312212i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 1.11i)T \)
good2 \( 1 + (1.64 - 2.05i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (3.33 + 1.60i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-1.47 + 1.84i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.990 - 1.24i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.263 + 1.15i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 4.37T + 19T^{2} \)
23 \( 1 + (-1.93 - 8.46i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-0.443 + 1.94i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 1.08T + 31T^{2} \)
37 \( 1 + (0.0740 - 0.324i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-3.96 - 1.91i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.93 - 1.41i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-4.93 + 6.19i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.847 - 3.71i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-8.45 + 4.07i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.964 - 4.22i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 1.74T + 67T^{2} \)
71 \( 1 + (-1.49 - 6.53i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (3.58 + 4.48i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (7.79 + 9.77i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-3.76 - 4.71i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 5.40T + 97T^{2} \)
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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

−11.36577850635572178966850839855, −9.923667253170402098256051241371, −9.204788592570739813652382825238, −8.605918752383840324657995703309, −7.63870903761533645663066291184, −7.09426690486323139660062624224, −5.93212158254015427951734066474, −4.99972729314427577481396994649, −3.59239425586031594528221039320, −0.832437725305347766130269558699, 0.62400265287847216045786820859, 2.68829622736624329010455266159, 3.50929447645130682004534485699, 4.34541411447014350778482561991, 6.79636779965383694553119816148, 7.46523375735057524492024427736, 8.318936081306068435555771635722, 9.306931879053653187620537350569, 10.21642917617522832723559942068, 10.76308949906306576096944631559

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