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Properties

Label 2-21e2-441.47-c1-0-12
Degree $2$
Conductor $441$
Sign $-0.709 - 0.704i$
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  + (1.76 + 1.89i)2-s + (−1.24 − 1.20i)3-s + (−0.351 + 4.69i)4-s + (2.19 + 2.74i)5-s + (0.0907 − 4.48i)6-s + (−0.860 − 2.50i)7-s + (−5.48 + 4.37i)8-s + (0.103 + 2.99i)9-s + (−1.35 + 9.00i)10-s + (−2.88 + 0.658i)11-s + (6.08 − 5.42i)12-s + (3.62 + 3.90i)13-s + (3.23 − 6.04i)14-s + (0.577 − 6.06i)15-s + (−8.64 − 1.30i)16-s + (−0.169 − 2.26i)17-s + ⋯
L(s)  = 1  + (1.24 + 1.34i)2-s + (−0.719 − 0.694i)3-s + (−0.175 + 2.34i)4-s + (0.980 + 1.22i)5-s + (0.0370 − 1.83i)6-s + (−0.325 − 0.945i)7-s + (−1.93 + 1.54i)8-s + (0.0343 + 0.999i)9-s + (−0.429 + 2.84i)10-s + (−0.869 + 0.198i)11-s + (1.75 − 1.56i)12-s + (1.00 + 1.08i)13-s + (0.864 − 1.61i)14-s + (0.149 − 1.56i)15-s + (−2.16 − 0.325i)16-s + (−0.0412 − 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.709 - 0.704i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.709 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848980 + 2.06116i\)
\(L(\frac12)\) \(\approx\) \(0.848980 + 2.06116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 + 1.20i)T \)
7 \( 1 + (0.860 + 2.50i)T \)
good2 \( 1 + (-1.76 - 1.89i)T + (-0.149 + 1.99i)T^{2} \)
5 \( 1 + (-2.19 - 2.74i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (2.88 - 0.658i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-3.62 - 3.90i)T + (-0.971 + 12.9i)T^{2} \)
17 \( 1 + (0.169 + 2.26i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (-0.908 - 0.524i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.819 + 1.70i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (4.64 + 6.81i)T + (-10.5 + 26.9i)T^{2} \)
31 \( 1 + (-4.78 - 2.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.52 + 1.72i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (0.380 - 0.969i)T + (-30.0 - 27.8i)T^{2} \)
43 \( 1 + (-1.79 - 4.56i)T + (-31.5 + 29.2i)T^{2} \)
47 \( 1 + (-6.63 + 6.15i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-1.60 + 2.35i)T + (-19.3 - 49.3i)T^{2} \)
59 \( 1 + (4.43 + 11.2i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (-15.4 + 1.15i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-5.16 + 8.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.67 - 3.48i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (0.764 + 2.47i)T + (-60.3 + 41.1i)T^{2} \)
79 \( 1 + (0.913 + 1.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.33 - 5.87i)T + (6.20 + 82.7i)T^{2} \)
89 \( 1 + (0.457 + 0.424i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 + (-2.24 - 1.29i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.58808119244037285162618685457, −10.81043674081948456226014985962, −9.773143290687551910241837639963, −8.063333950150752461233005368070, −7.22776263276183129288588858308, −6.57341975840957405392956720639, −6.12071040592063989892297150466, −5.08560183389929039948780635002, −3.85769777849243265303168570711, −2.44374843003868065727538943070, 1.12993197204114923073853286700, 2.66499407005301802546075427570, 3.86632601785403929456065914336, 5.09576258807904366176983549005, 5.59899421274520464887525862635, 6.02836685021097507689394931612, 8.589760754472070905443576386635, 9.411253290512011395608216684316, 10.21195342278868390916845127359, 10.85427298735224934439974551131

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