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Properties

Label 2-21e2-63.59-c1-0-8
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: $\chi_{441} (374, \cdot )$
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

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

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