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Properties

Label 2-21e2-49.43-c1-0-1
Degree $2$
Conductor $441$
Sign $-0.989 + 0.143i$
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  + (−0.494 + 0.620i)2-s + (0.304 + 1.33i)4-s + (−1.66 − 0.803i)5-s + (0.814 − 2.51i)7-s + (−2.41 − 1.16i)8-s + (1.32 − 0.637i)10-s + (−2.05 + 2.57i)11-s + (−3.75 + 4.70i)13-s + (1.15 + 1.75i)14-s + (−0.555 + 0.267i)16-s + (−1.06 + 4.68i)17-s − 3.59·19-s + (0.564 − 2.47i)20-s + (−0.582 − 2.55i)22-s + (−0.399 − 1.74i)23-s + ⋯
L(s)  = 1  + (−0.349 + 0.438i)2-s + (0.152 + 0.667i)4-s + (−0.746 − 0.359i)5-s + (0.307 − 0.951i)7-s + (−0.852 − 0.410i)8-s + (0.418 − 0.201i)10-s + (−0.620 + 0.777i)11-s + (−1.04 + 1.30i)13-s + (0.309 + 0.468i)14-s + (−0.138 + 0.0668i)16-s + (−0.259 + 1.13i)17-s − 0.825·19-s + (0.126 − 0.552i)20-s + (−0.124 − 0.544i)22-s + (−0.0832 − 0.364i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.989 + 0.143i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.989 + 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0222503 - 0.309373i\)
\(L(\frac12)\) \(\approx\) \(0.0222503 - 0.309373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.814 + 2.51i)T \)
good2 \( 1 + (0.494 - 0.620i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (1.66 + 0.803i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.05 - 2.57i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.75 - 4.70i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.06 - 4.68i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 3.59T + 19T^{2} \)
23 \( 1 + (0.399 + 1.74i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-1.56 + 6.84i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 7.52T + 31T^{2} \)
37 \( 1 + (0.545 - 2.39i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-5.35 - 2.57i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-6.85 + 3.30i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.03 - 6.31i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.61 - 11.4i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-0.864 + 0.416i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.40 - 6.17i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 7.63T + 67T^{2} \)
71 \( 1 + (0.688 + 3.01i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.29 - 1.62i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 7.73T + 79T^{2} \)
83 \( 1 + (1.26 + 1.58i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (3.01 + 3.78i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−11.65888058162507661965825443732, −10.70332643325071681211056773295, −9.668843906801237685520380111859, −8.651155786181530438165127084984, −7.76763701503543379494249345083, −7.30777003683154077493427399906, −6.29682714729519049292627909014, −4.42809119752743559696543711476, −4.10539420289541637978051034848, −2.25653426624936024630258690018, 0.20123426619303135316578219902, 2.30038776444457249822479588263, 3.21709150097426926811061980706, 5.12102124543823402877310959414, 5.63898094529495272942928477993, 7.04822666678438460461137806938, 8.031014861906950351703491898603, 8.931163443077120838799493050859, 9.827690150099716033617445688432, 10.88557404738322535454038445125

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