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Properties

Label 2-21e2-147.104-c1-0-17
Degree $2$
Conductor $441$
Sign $0.314 + 0.949i$
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  + (0.920 + 0.733i)2-s + (−0.136 − 0.599i)4-s + (−1.46 − 0.703i)5-s + (−2.01 − 1.70i)7-s + (1.33 − 2.77i)8-s + (−0.828 − 1.71i)10-s + (2.35 + 1.87i)11-s + (−2.44 − 1.94i)13-s + (−0.604 − 3.05i)14-s + (2.15 − 1.03i)16-s + (0.882 − 3.86i)17-s − 6.34i·19-s + (−0.221 + 0.971i)20-s + (0.788 + 3.45i)22-s + (−1.23 + 0.280i)23-s + ⋯
L(s)  = 1  + (0.650 + 0.518i)2-s + (−0.0683 − 0.299i)4-s + (−0.653 − 0.314i)5-s + (−0.763 − 0.646i)7-s + (0.472 − 0.980i)8-s + (−0.261 − 0.543i)10-s + (0.710 + 0.566i)11-s + (−0.676 − 0.539i)13-s + (−0.161 − 0.816i)14-s + (0.538 − 0.259i)16-s + (0.214 − 0.938i)17-s − 1.45i·19-s + (−0.0495 + 0.217i)20-s + (0.168 + 0.736i)22-s + (−0.256 + 0.0585i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13261 - 0.817872i\)
\(L(\frac12)\) \(\approx\) \(1.13261 - 0.817872i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 + 1.70i)T \)
good2 \( 1 + (-0.920 - 0.733i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (1.46 + 0.703i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.35 - 1.87i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.44 + 1.94i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.882 + 3.86i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.34iT - 19T^{2} \)
23 \( 1 + (1.23 - 0.280i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-7.25 - 1.65i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 1.49iT - 31T^{2} \)
37 \( 1 + (1.72 - 7.54i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-9.09 - 4.37i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.978 - 0.471i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.44 - 6.82i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.104 + 0.0237i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-1.54 + 0.746i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (10.5 + 2.41i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + (-3.29 + 0.752i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-8.80 + 7.01i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + (-2.15 - 2.70i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (7.65 + 9.59i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 6.40iT - 97T^{2} \)
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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

−10.94725700328669836626527589931, −9.872161602257389058423179090739, −9.381120251262221929425318705327, −7.907187674067313011104847938987, −6.98704248133570764054726150235, −6.39243601411875325580686009240, −4.91286924560692231307331087209, −4.41020756518035747192019371483, −3.08712397342200778867606489519, −0.73238562712799389994606876612, 2.21323002298309078761805105029, 3.52375874735145277981993751073, 4.02531896159181264680170218052, 5.51729262157858447648681930294, 6.50234514999413512776542555852, 7.73076747097530754639348583052, 8.516919542158099924087521834031, 9.559628697218123490776604914881, 10.64184807122341690965725009952, 11.60176301517847929530904281693

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