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Properties

Label 2-21e2-21.17-c1-0-13
Degree $2$
Conductor $441$
Sign $-0.496 + 0.867i$
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  + (2.09 − 1.20i)2-s + (1.91 − 3.31i)4-s + (−1.68 − 2.92i)5-s − 4.41i·8-s + (−7.06 − 4.07i)10-s + (−0.717 − 0.414i)11-s + 3.37i·13-s + (−1.49 − 2.59i)16-s + (−0.699 + 1.21i)17-s + (5.85 − 3.37i)19-s − 12.9·20-s − 2·22-s + (1.73 − i)23-s + (−3.20 + 5.55i)25-s + (4.07 + 7.06i)26-s + ⋯
L(s)  = 1  + (1.47 − 0.853i)2-s + (0.957 − 1.65i)4-s + (−0.755 − 1.30i)5-s − 1.56i·8-s + (−2.23 − 1.28i)10-s + (−0.216 − 0.124i)11-s + 0.937i·13-s + (−0.374 − 0.649i)16-s + (−0.169 + 0.293i)17-s + (1.34 − 0.775i)19-s − 2.89·20-s − 0.426·22-s + (0.361 − 0.208i)23-s + (−0.641 + 1.11i)25-s + (0.799 + 1.38i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.496 + 0.867i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.496 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35571 - 2.33752i\)
\(L(\frac12)\) \(\approx\) \(1.35571 - 2.33752i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-2.09 + 1.20i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.68 + 2.92i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.717 + 0.414i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 + (0.699 - 1.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.85 + 3.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 + (-5.85 - 3.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.29 - 2.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.15T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + (3.37 + 5.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.12 - 3.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.37 - 5.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 - 4.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.24 - 7.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.82iT - 71T^{2} \)
73 \( 1 + (-1.21 - 0.699i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.82 - 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.39iT - 97T^{2} \)
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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

−11.45854520596263105303037744058, −10.19298276487069841118120966418, −9.112084232966082495253495086101, −8.156613909334367431826250926420, −6.83141620964273971347623156850, −5.55217869687147994826987086070, −4.70825109903186298315118607293, −4.10466847774671053048452115731, −2.83515132151366594540984831671, −1.23142527723345827614485882521, 2.95269210727867407935767854447, 3.46855220359586757240790971847, 4.73867512997494875203259633989, 5.74951018022103172243552840556, 6.68089603231957541078914612528, 7.50793244286389378584666699973, 8.003438329387267025155774498810, 9.782553525081247947762093689575, 10.84566695097665190468682543048, 11.65120042078781166812545321847

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