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Properties

Label 2-21e2-147.104-c1-0-4
Degree $2$
Conductor $441$
Sign $-0.828 - 0.560i$
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  + (2.12 + 1.69i)2-s + (1.19 + 5.25i)4-s + (−2.70 − 1.30i)5-s + (−0.967 + 2.46i)7-s + (−3.99 + 8.30i)8-s + (−3.53 − 7.34i)10-s + (1.87 + 1.49i)11-s + (1.93 + 1.54i)13-s + (−6.23 + 3.59i)14-s + (−12.8 + 6.19i)16-s + (1.15 − 5.04i)17-s + 0.270i·19-s + (3.59 − 15.7i)20-s + (1.44 + 6.34i)22-s + (8.11 − 1.85i)23-s + ⋯
L(s)  = 1  + (1.50 + 1.19i)2-s + (0.599 + 2.62i)4-s + (−1.20 − 0.582i)5-s + (−0.365 + 0.930i)7-s + (−1.41 + 2.93i)8-s + (−1.11 − 2.32i)10-s + (0.564 + 0.449i)11-s + (0.536 + 0.427i)13-s + (−1.66 + 0.960i)14-s + (−3.21 + 1.54i)16-s + (0.279 − 1.22i)17-s + 0.0621i·19-s + (0.804 − 3.52i)20-s + (0.308 + 1.35i)22-s + (1.69 − 0.386i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.828 - 0.560i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722124 + 2.35555i\)
\(L(\frac12)\) \(\approx\) \(0.722124 + 2.35555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.967 - 2.46i)T \)
good2 \( 1 + (-2.12 - 1.69i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (2.70 + 1.30i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-1.87 - 1.49i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.93 - 1.54i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.15 + 5.04i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 0.270iT - 19T^{2} \)
23 \( 1 + (-8.11 + 1.85i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.64 + 0.603i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 8.55iT - 31T^{2} \)
37 \( 1 + (-1.31 + 5.77i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-0.862 - 0.415i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-5.37 + 2.58i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-5.40 + 6.77i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.39 - 0.317i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (5.63 - 2.71i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-12.1 - 2.76i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.18T + 67T^{2} \)
71 \( 1 + (3.43 - 0.784i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.58 - 5.25i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 8.56T + 79T^{2} \)
83 \( 1 + (0.701 + 0.879i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (10.3 + 13.0i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−11.96208438808059318278154955346, −11.22128209743267763568326896064, −9.059687031015272350681906740373, −8.614363407911821308267662689794, −7.38337584747206720394733159156, −6.86729517415308118713803753562, −5.64313903133509393149245798573, −4.81589460494102150390529781528, −3.93625053122140348178401965844, −2.90496063717483385299742002690, 1.07199754965571532454266616859, 3.04679022354980638336650883305, 3.71352494047184431391842052965, 4.36724008057258637603940692758, 5.82002308216411454350644296047, 6.69558113945384986782291918349, 7.77935003123671703346651975557, 9.357639712566679380901577817045, 10.48881803554988418595123188530, 11.06403710372247753370305826387

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