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Properties

Label 2-21e2-63.47-c1-0-11
Degree $2$
Conductor $441$
Sign $0.729 - 0.684i$
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  + (−2.23 + 1.28i)2-s + (−0.625 − 1.61i)3-s + (2.32 − 4.02i)4-s + 2.33·5-s + (3.47 + 2.80i)6-s + 6.82i·8-s + (−2.21 + 2.01i)9-s + (−5.20 + 3.00i)10-s + 4.36i·11-s + (−7.95 − 1.23i)12-s + (1.14 − 0.660i)13-s + (−1.45 − 3.76i)15-s + (−4.15 − 7.18i)16-s + (2.89 + 5.01i)17-s + (2.34 − 7.36i)18-s + (0.584 + 0.337i)19-s + ⋯
L(s)  = 1  + (−1.57 + 0.911i)2-s + (−0.360 − 0.932i)3-s + (1.16 − 2.01i)4-s + 1.04·5-s + (1.41 + 1.14i)6-s + 2.41i·8-s + (−0.739 + 0.673i)9-s + (−1.64 + 0.950i)10-s + 1.31i·11-s + (−2.29 − 0.357i)12-s + (0.317 − 0.183i)13-s + (−0.376 − 0.972i)15-s + (−1.03 − 1.79i)16-s + (0.701 + 1.21i)17-s + (0.553 − 1.73i)18-s + (0.134 + 0.0774i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.729 - 0.684i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633827 + 0.250820i\)
\(L(\frac12)\) \(\approx\) \(0.633827 + 0.250820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.625 + 1.61i)T \)
7 \( 1 \)
good2 \( 1 + (2.23 - 1.28i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.33T + 5T^{2} \)
11 \( 1 - 4.36iT - 11T^{2} \)
13 \( 1 + (-1.14 + 0.660i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.89 - 5.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.584 - 0.337i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.56iT - 23T^{2} \)
29 \( 1 + (-3.86 - 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.47 - 2.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.50 - 2.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.29 - 5.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.89 + 6.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.246 + 0.427i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.59 - 2.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.15 - 3.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.77 + 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 + (-13.0 + 7.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.30 - 9.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.32 + 9.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.66 + 2.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.7 - 7.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63946937054786435575567910852, −10.26604898157232670048843311844, −9.309729811355003398369480392374, −8.336357597261314861475342761542, −7.64950832332878882990211804990, −6.58755559080042918039708996499, −6.17834708542315595784194135837, −5.10964887653535842199350421416, −2.22222509473179928936183645209, −1.25602008257790060964032722648, 0.890744871942410308073399058466, 2.61610983142804641786705263621, 3.59815574192345450103531188074, 5.37050918735169608243362923233, 6.34816772650216776757558324417, 7.77494682222552945830394372787, 8.774629507499495721308021485485, 9.526876020534024715377455616494, 9.887851028757316886019641060145, 10.90370871560872057481560084448

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