Properties

Label 2-21e2-441.122-c1-0-48
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: $\chi_{441} (122, \cdot )$
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

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

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