Properties

Label 2-21e2-63.16-c1-0-3
Degree $2$
Conductor $441$
Sign $0.978 - 0.203i$
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  + (−1.08 − 1.88i)2-s + (−1.18 − 1.26i)3-s + (−1.36 + 2.36i)4-s + 1.26·5-s + (−1.10 + 3.60i)6-s + 1.60·8-s + (−0.213 + 2.99i)9-s + (−1.38 − 2.39i)10-s − 5.47·11-s + (4.61 − 1.06i)12-s + (2.37 + 4.10i)13-s + (−1.49 − 1.60i)15-s + (0.992 + 1.71i)16-s + (2.40 + 4.17i)17-s + (5.87 − 2.85i)18-s + (−2.69 + 4.66i)19-s + ⋯
L(s)  = 1  + (−0.769 − 1.33i)2-s + (−0.681 − 0.731i)3-s + (−0.684 + 1.18i)4-s + 0.567·5-s + (−0.450 + 1.47i)6-s + 0.566·8-s + (−0.0710 + 0.997i)9-s + (−0.436 − 0.755i)10-s − 1.65·11-s + (1.33 − 0.306i)12-s + (0.658 + 1.13i)13-s + (−0.386 − 0.415i)15-s + (0.248 + 0.429i)16-s + (0.584 + 1.01i)17-s + (1.38 − 0.672i)18-s + (−0.617 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.978 - 0.203i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.978 - 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338433 + 0.0348713i\)
\(L(\frac12)\) \(\approx\) \(0.338433 + 0.0348713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 + 1.26i)T \)
7 \( 1 \)
good2 \( 1 + (1.08 + 1.88i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.26T + 5T^{2} \)
11 \( 1 + 5.47T + 11T^{2} \)
13 \( 1 + (-2.37 - 4.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.40 - 4.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.17T + 23T^{2} \)
29 \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.732 + 1.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.959 - 1.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.94 + 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.57 + 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.57 - 6.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.154 - 0.267i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.17 + 8.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + (-5.27 - 9.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.08 - 8.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.59 - 4.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 - 4.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.02919256236801537653207039623, −10.34102074322142933524138303037, −9.818485561516065625991026543598, −8.371857782684098876156764470013, −7.924328814402410591057849485827, −6.33106133490507069052250995884, −5.62738476906453049076815240060, −3.99398859016555630594433223862, −2.35400973344042445191460778546, −1.59418872793484999466829615876, 0.29341811367025553024115455611, 3.02465802683078018685858506612, 4.93378747273956508979343497835, 5.55355912003341051778334008691, 6.29865797812318128121607291702, 7.44772138304508762502424740695, 8.298219301033532264775943120087, 9.228076578398264033636461259994, 10.17905430158408240267511750716, 10.53348402864157743672061121571

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