Properties

Label 2-21e2-49.39-c1-0-12
Degree $2$
Conductor $441$
Sign $0.704 - 0.709i$
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  + (2.25 + 0.339i)2-s + (3.05 + 0.943i)4-s + (0.260 + 3.47i)5-s + (1.30 − 2.30i)7-s + (2.46 + 1.18i)8-s + (−0.594 + 7.92i)10-s + (−0.728 − 1.85i)11-s + (−0.725 + 0.909i)13-s + (3.71 − 4.75i)14-s + (−0.128 − 0.0873i)16-s + (−0.130 − 0.120i)17-s + (2.36 + 4.09i)19-s + (−2.48 + 10.8i)20-s + (−1.01 − 4.43i)22-s + (−2.75 + 2.55i)23-s + ⋯
L(s)  = 1  + (1.59 + 0.240i)2-s + (1.52 + 0.471i)4-s + (0.116 + 1.55i)5-s + (0.491 − 0.870i)7-s + (0.872 + 0.420i)8-s + (−0.187 + 2.50i)10-s + (−0.219 − 0.559i)11-s + (−0.201 + 0.252i)13-s + (0.993 − 1.27i)14-s + (−0.0320 − 0.0218i)16-s + (−0.0315 − 0.0293i)17-s + (0.542 + 0.940i)19-s + (−0.555 + 2.43i)20-s + (−0.215 − 0.945i)22-s + (−0.574 + 0.533i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.04679 + 1.26850i\)
\(L(\frac12)\) \(\approx\) \(3.04679 + 1.26850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.30 + 2.30i)T \)
good2 \( 1 + (-2.25 - 0.339i)T + (1.91 + 0.589i)T^{2} \)
5 \( 1 + (-0.260 - 3.47i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (0.728 + 1.85i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (0.725 - 0.909i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.130 + 0.120i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.75 - 2.55i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-2.03 + 8.90i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-4.06 + 7.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.87 - 1.50i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (4.57 + 2.20i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.63 + 0.787i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.31 - 0.198i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-7.14 - 2.20i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.0395 - 0.528i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (3.38 - 1.04i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.223 + 0.387i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.323 - 1.41i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (13.2 - 1.99i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.35 - 4.21i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (2.62 - 6.67i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 - 1.58T + 97T^{2} \)
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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

−11.52857851023119673642490284930, −10.58083207548779520709854016129, −9.854525903554481710593410952384, −7.962188612651495326493620754302, −7.23852110695276773325721404981, −6.35497363561317083810190638654, −5.60406753846529712260375950121, −4.23690651641089626849899433316, −3.48677814599460388787339307153, −2.37237856431447303346685683614, 1.70771508725620660031871561533, 3.01029843594140223642648314854, 4.59484080199832722731252714088, 4.97571404953387541410120467996, 5.71666227034278127550017022827, 7.00853469652196018289901676377, 8.456926355581134592361090014084, 9.022084761370605200272598772859, 10.33790623994974556941980585407, 11.52338744319048529460749733062

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