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Properties

Label 2-21e2-63.47-c1-0-34
Degree $2$
Conductor $441$
Sign $-0.628 + 0.778i$
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  + (2.34 − 1.35i)2-s + (−1.12 − 1.31i)3-s + (2.66 − 4.62i)4-s + 1.20·5-s + (−4.42 − 1.54i)6-s − 9.04i·8-s + (−0.447 + 2.96i)9-s + (2.82 − 1.62i)10-s + 2.48i·11-s + (−9.08 + 1.71i)12-s + (−1.63 + 0.942i)13-s + (−1.35 − 1.57i)15-s + (−6.90 − 11.9i)16-s + (−0.601 − 1.04i)17-s + (2.96 + 7.56i)18-s + (6.46 + 3.73i)19-s + ⋯
L(s)  = 1  + (1.65 − 0.957i)2-s + (−0.652 − 0.757i)3-s + (1.33 − 2.31i)4-s + 0.537·5-s + (−1.80 − 0.632i)6-s − 3.19i·8-s + (−0.149 + 0.988i)9-s + (0.892 − 0.515i)10-s + 0.749i·11-s + (−2.62 + 0.496i)12-s + (−0.452 + 0.261i)13-s + (−0.350 − 0.407i)15-s + (−1.72 − 2.99i)16-s + (−0.145 − 0.252i)17-s + (0.699 + 1.78i)18-s + (1.48 + 0.856i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.628 + 0.778i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.628 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29788 - 2.71607i\)
\(L(\frac12)\) \(\approx\) \(1.29788 - 2.71607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 + 1.31i)T \)
7 \( 1 \)
good2 \( 1 + (-2.34 + 1.35i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.20T + 5T^{2} \)
11 \( 1 - 2.48iT - 11T^{2} \)
13 \( 1 + (1.63 - 0.942i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.04iT - 23T^{2} \)
29 \( 1 + (0.173 + 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.03 + 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.865 - 1.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.36 + 5.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00656 + 0.0113i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.717 - 1.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.58 + 4.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.10 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.73 + 5.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 4.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (-7.51 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.74 + 4.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.60 - 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.98 - 6.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.06 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.25504226696228038978586722552, −10.20701245227249464237980031364, −9.600289613487631184355935072116, −7.51940862911578729757686276459, −6.70746521643628610036610487735, −5.59874631178188113100292015874, −5.18770951763638669128871175004, −3.87864822504427530211702193219, −2.42661672843977869980268213378, −1.49575350614502306048352638216, 2.85722998468401992150466778001, 3.86243876711060001996191167101, 5.00257402747004945238011931131, 5.55026960803174977754738508163, 6.38556700498151544115185727202, 7.29219494680397472713579694620, 8.521518870395994380142972255447, 9.694819383789680100394634232016, 10.92883973604325104465870381297, 11.64701448336119475866082051165

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