Properties

Label 2-21e2-63.25-c1-0-10
Degree $2$
Conductor $441$
Sign $0.112 - 0.993i$
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  − 0.239·2-s + (1.09 + 1.34i)3-s − 1.94·4-s + (0.590 − 1.02i)5-s + (−0.260 − 0.321i)6-s + 0.942·8-s + (−0.619 + 2.93i)9-s + (−0.141 + 0.244i)10-s + (1.85 + 3.20i)11-s + (−2.11 − 2.61i)12-s + (0.5 + 0.866i)13-s + (2.02 − 0.321i)15-s + 3.66·16-s + (−3.47 + 6.01i)17-s + (0.148 − 0.701i)18-s + (0.971 + 1.68i)19-s + ⋯
L(s)  = 1  − 0.169·2-s + (0.629 + 0.776i)3-s − 0.971·4-s + (0.264 − 0.457i)5-s + (−0.106 − 0.131i)6-s + 0.333·8-s + (−0.206 + 0.978i)9-s + (−0.0446 + 0.0774i)10-s + (0.558 + 0.967i)11-s + (−0.611 − 0.754i)12-s + (0.138 + 0.240i)13-s + (0.522 − 0.0830i)15-s + 0.915·16-s + (−0.841 + 1.45i)17-s + (0.0349 − 0.165i)18-s + (0.222 + 0.385i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.944231 + 0.843602i\)
\(L(\frac12)\) \(\approx\) \(0.944231 + 0.843602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.09 - 1.34i)T \)
7 \( 1 \)
good2 \( 1 + 0.239T + 2T^{2} \)
5 \( 1 + (-0.590 + 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.85 - 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.47 - 6.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.971 - 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.80 + 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.119 - 0.207i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 + (-4.77 - 8.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.09 + 8.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.11 - 1.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 + (-5.80 + 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.60T + 59T^{2} \)
61 \( 1 - 7.60T + 61T^{2} \)
67 \( 1 - 3.50T + 67T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 + (-7.57 + 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.37T + 79T^{2} \)
83 \( 1 + (3.47 - 6.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.37 - 2.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.58 + 6.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.03983816277787099060980677099, −10.12392992174525061398172569478, −9.485837872083985142891719841580, −8.706570953738336272696781125344, −8.162664048376671333225334130436, −6.71805888634145325840861049414, −5.24434896010162046004734743416, −4.46850391429830045957973243690, −3.63589354217713771305739119544, −1.80334589498930851154051725232, 0.873394054344896180135246586070, 2.70002845892550777932718434439, 3.75760701712055102898928713463, 5.19415420416481064096068118927, 6.38091920192777813985259009833, 7.30124706600830106042133006169, 8.291396895544287323128136860353, 9.117339039489341182055169137362, 9.616800903838214841710037955482, 10.98778650013616277055452982355

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