Properties

Label 2-21e2-49.8-c1-0-22
Degree $2$
Conductor $441$
Sign $-0.142 + 0.989i$
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  + (0.309 + 0.387i)2-s + (0.390 − 1.70i)4-s + (−0.734 + 0.353i)5-s + (−2.35 − 1.20i)7-s + (1.67 − 0.807i)8-s + (−0.364 − 0.175i)10-s + (−2.98 − 3.74i)11-s + (−1.29 − 1.62i)13-s + (−0.262 − 1.28i)14-s + (−2.32 − 1.12i)16-s + (−0.242 − 1.06i)17-s + 7.79·19-s + (0.318 + 1.39i)20-s + (0.528 − 2.31i)22-s + (−0.652 + 2.85i)23-s + ⋯
L(s)  = 1  + (0.218 + 0.274i)2-s + (0.195 − 0.854i)4-s + (−0.328 + 0.158i)5-s + (−0.890 − 0.454i)7-s + (0.593 − 0.285i)8-s + (−0.115 − 0.0554i)10-s + (−0.900 − 1.12i)11-s + (−0.360 − 0.451i)13-s + (−0.0700 − 0.343i)14-s + (−0.582 − 0.280i)16-s + (−0.0588 − 0.257i)17-s + 1.78·19-s + (0.0711 + 0.311i)20-s + (0.112 − 0.493i)22-s + (−0.136 + 0.595i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.142 + 0.989i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.142 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729401 - 0.841768i\)
\(L(\frac12)\) \(\approx\) \(0.729401 - 0.841768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.35 + 1.20i)T \)
good2 \( 1 + (-0.309 - 0.387i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (0.734 - 0.353i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.98 + 3.74i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.29 + 1.62i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.242 + 1.06i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 7.79T + 19T^{2} \)
23 \( 1 + (0.652 - 2.85i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.37 + 10.4i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + 3.31T + 31T^{2} \)
37 \( 1 + (0.439 + 1.92i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-7.26 + 3.49i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-5.89 - 2.83i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-5.16 - 6.48i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.782 + 3.42i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-7.50 - 3.61i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (1.32 + 5.81i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 - 1.24T + 67T^{2} \)
71 \( 1 + (1.67 - 7.35i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.69 + 4.63i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.49T + 79T^{2} \)
83 \( 1 + (6.02 - 7.55i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (3.82 - 4.79i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−10.93101324525899762898724828123, −9.901325055637454233020402502813, −9.396485778922029021713408724639, −7.73987471877483977117159480277, −7.28168097139441411164272989426, −5.88564105938730570517964634063, −5.48267286071778767783015265716, −3.92714936227898870779693046709, −2.76203738324108432978804428925, −0.63466638665682594874347240552, 2.26510522334993930172279773467, 3.28646845547595157473638726001, 4.42629118180646947317978684057, 5.52639616019509330490517470152, 7.04383956625643461580863702764, 7.50577581157236507783930929916, 8.670173100657471727756673439494, 9.589493783755470602009063102486, 10.50366013631371537717653732757, 11.61589764873410376456297814206

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