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Properties

Label 2-21e2-63.47-c1-0-35
Degree $2$
Conductor $441$
Sign $-0.839 + 0.543i$
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  + (2.05 − 1.18i)2-s + (0.371 − 1.69i)3-s + (1.81 − 3.14i)4-s − 3.43·5-s + (−1.24 − 3.91i)6-s − 3.86i·8-s + (−2.72 − 1.25i)9-s + (−7.05 + 4.07i)10-s − 0.313i·11-s + (−4.64 − 4.23i)12-s + (5.09 − 2.94i)13-s + (−1.27 + 5.81i)15-s + (−0.958 − 1.65i)16-s + (0.476 + 0.825i)17-s + (−7.08 + 0.644i)18-s + (1.09 + 0.630i)19-s + ⋯
L(s)  = 1  + (1.45 − 0.838i)2-s + (0.214 − 0.976i)3-s + (0.907 − 1.57i)4-s − 1.53·5-s + (−0.507 − 1.59i)6-s − 1.36i·8-s + (−0.907 − 0.419i)9-s + (−2.23 + 1.28i)10-s − 0.0946i·11-s + (−1.34 − 1.22i)12-s + (1.41 − 0.816i)13-s + (−0.329 + 1.50i)15-s + (−0.239 − 0.414i)16-s + (0.115 + 0.200i)17-s + (−1.67 + 0.152i)18-s + (0.250 + 0.144i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.839 + 0.543i$
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: \((2,\ 441,\ (\ :1/2),\ -0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713132 - 2.41457i\)
\(L(\frac12)\) \(\approx\) \(0.713132 - 2.41457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.371 + 1.69i)T \)
7 \( 1 \)
good2 \( 1 + (-2.05 + 1.18i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.43T + 5T^{2} \)
11 \( 1 + 0.313iT - 11T^{2} \)
13 \( 1 + (-5.09 + 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.476 - 0.825i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 0.630i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.82iT - 23T^{2} \)
29 \( 1 + (-3.43 - 1.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.53 - 2.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.68 - 4.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0699 + 0.121i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.00 - 1.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.3 - 5.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.824 - 1.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.57 + 1.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.934 + 1.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-0.354 + 0.204i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.23 + 9.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.00 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.05 - 1.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.12204723307662261081187408280, −10.52669665910271836613315459226, −8.549523658719174236137555428594, −8.061688469252297653912096701255, −6.78130915935653040364403581158, −5.91710009167501637808876256584, −4.62838089717568510498780984743, −3.56640609342516492603028427445, −2.88443039118741297059560945156, −1.09964606497185778560954786002, 3.19069585189875307143607349780, 3.89396981164372724562849451112, 4.50999073655306493369736304210, 5.57903802008415210548630776112, 6.67587650148134868329693449639, 7.73598012103679202809411631554, 8.420556014183185632803531087438, 9.587186488196466760740993819337, 11.11784980570480728696931164021, 11.51529931613400017089524557359

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