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Properties

Label 2-21e2-49.8-c1-0-21
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: $\chi_{441} (253, \cdot )$
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

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

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