Properties

Label 2-21e2-441.142-c1-0-39
Degree $2$
Conductor $441$
Sign $0.914 - 0.405i$
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  + (−0.874 + 2.22i)2-s + (0.970 − 1.43i)3-s + (−2.73 − 2.53i)4-s + (3.39 − 1.63i)5-s + (2.34 + 3.41i)6-s + (1.06 + 2.42i)7-s + (3.72 − 1.79i)8-s + (−1.11 − 2.78i)9-s + (0.674 + 9.00i)10-s + (−2.04 − 2.56i)11-s + (−6.29 + 1.45i)12-s + (2.32 − 5.92i)13-s + (−6.32 + 0.262i)14-s + (0.952 − 6.46i)15-s + (0.183 + 2.44i)16-s + (−1.49 + 1.38i)17-s + ⋯
L(s)  = 1  + (−0.618 + 1.57i)2-s + (0.560 − 0.828i)3-s + (−1.36 − 1.26i)4-s + (1.52 − 0.732i)5-s + (0.958 + 1.39i)6-s + (0.403 + 0.914i)7-s + (1.31 − 0.635i)8-s + (−0.371 − 0.928i)9-s + (0.213 + 2.84i)10-s + (−0.617 − 0.774i)11-s + (−1.81 + 0.421i)12-s + (0.644 − 1.64i)13-s + (−1.69 + 0.0700i)14-s + (0.245 − 1.66i)15-s + (0.0457 + 0.610i)16-s + (−0.361 + 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.914 - 0.405i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40761 + 0.298038i\)
\(L(\frac12)\) \(\approx\) \(1.40761 + 0.298038i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.970 + 1.43i)T \)
7 \( 1 + (-1.06 - 2.42i)T \)
good2 \( 1 + (0.874 - 2.22i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-3.39 + 1.63i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.04 + 2.56i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.32 + 5.92i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (1.49 - 1.38i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.0966 - 0.167i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.854 - 3.74i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.654 + 0.201i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (2.46 - 4.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.59 - 2.96i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-2.42 - 1.65i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-6.13 + 4.17i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (3.31 - 8.45i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (8.97 - 2.76i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.05 + 0.719i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (3.51 - 3.26i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-2.89 + 5.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.83 - 8.05i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-15.7 - 2.37i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-2.64 - 4.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.110 - 0.282i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.0719 - 0.183i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.137 + 0.238i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92905456880019619912270904459, −9.637018781513049560282079694666, −9.053537629441065500111714434026, −8.240360880573634853027996070241, −7.86281325180971378624788273886, −6.29870429267833720031456488211, −5.82155495007385573655583840995, −5.23539820960577139551939578983, −2.73149485128768721304845091573, −1.17542465722559851589882831091, 1.87702113992096196040418345886, 2.48215835376684971118447299841, 3.86902087609916249712543799890, 4.74311441779352524132404487403, 6.41502311578608331596389956486, 7.73152988412399885683168504407, 9.004557706164686707045891089141, 9.580287118757335609989179111693, 10.12445842568637130298544492323, 10.94894245072027140533160740748

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