Properties

Label 2-21e2-147.41-c1-0-1
Degree $2$
Conductor $441$
Sign $-0.999 + 0.0198i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0198i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.999 + 0.0198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00444918 - 0.447985i\)
\(L(\frac12)\) \(\approx\) \(0.00444918 - 0.447985i\)
\(L(\frac{3}{2})\) 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

−11.72325723875655263487841104272, −10.24036937453144759145156365096, −9.441612669297400074589052049997, −9.130218012039882114174776265313, −7.85704028987633559105069906194, −7.13724349519274850869013840386, −6.07255393004165609124111616768, −5.09973974674906457927412267606, −3.56991201274485740567842163224, −2.18713574968892956486618996177, 0.32297536722017209841099851023, 2.14503407282848752967586183448, 3.25473790716624353559720578468, 4.98158236194811280928560731991, 5.95471454525985609822662973260, 6.92029699115032819595445813053, 8.140509488658543322443497993740, 9.133860566410163254312556141562, 9.953311531857598544244184980666, 10.50884302608201003058006677143

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