Properties

Label 2-21e2-63.4-c1-0-21
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: $\chi_{441} (67, \cdot )$
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

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

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