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Properties

Label 2-21e2-441.130-c1-0-33
Degree $2$
Conductor $441$
Sign $0.999 - 0.00842i$
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  + (1.19 − 0.813i)2-s + (1.71 + 0.210i)3-s + (0.0320 − 0.0815i)4-s + (−0.000507 − 0.00222i)5-s + (2.22 − 1.14i)6-s + (0.314 + 2.62i)7-s + (0.614 + 2.69i)8-s + (2.91 + 0.725i)9-s + (−0.00241 − 0.00223i)10-s + (−2.93 − 1.41i)11-s + (0.0722 − 0.133i)12-s + (1.97 − 1.34i)13-s + (2.51 + 2.88i)14-s + (−0.000403 − 0.00392i)15-s + (3.05 + 2.83i)16-s + (0.0598 + 0.152i)17-s + ⋯
L(s)  = 1  + (0.844 − 0.575i)2-s + (0.992 + 0.121i)3-s + (0.0160 − 0.0407i)4-s + (−0.000226 − 0.000993i)5-s + (0.907 − 0.468i)6-s + (0.118 + 0.992i)7-s + (0.217 + 0.952i)8-s + (0.970 + 0.241i)9-s + (−0.000763 − 0.000708i)10-s + (−0.884 − 0.425i)11-s + (0.0208 − 0.0385i)12-s + (0.546 − 0.372i)13-s + (0.671 + 0.769i)14-s + (−0.000104 − 0.00101i)15-s + (0.763 + 0.708i)16-s + (0.0145 + 0.0369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 - 0.00842i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.999 - 0.00842i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79762 + 0.0117790i\)
\(L(\frac12)\) \(\approx\) \(2.79762 + 0.0117790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 - 0.210i)T \)
7 \( 1 + (-0.314 - 2.62i)T \)
good2 \( 1 + (-1.19 + 0.813i)T + (0.730 - 1.86i)T^{2} \)
5 \( 1 + (0.000507 + 0.00222i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (2.93 + 1.41i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.97 + 1.34i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (-0.0598 - 0.152i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (1.97 + 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.65 + 5.83i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (3.91 + 0.589i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (2.86 + 4.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.85 - 0.882i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-2.43 - 0.750i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (2.71 - 0.836i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-0.462 + 0.315i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (8.34 - 1.25i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-4.53 + 1.39i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-5.47 - 13.9i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (2.35 + 4.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.98 + 6.25i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.543 - 7.24i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-6.21 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.3 - 9.08i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (6.52 + 4.44i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-3.00 - 5.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.18124971318582447440446111823, −10.44085801894754140861987067764, −9.159043246707449054052916397511, −8.390713304704661427229754882725, −7.82137303544940010872546796334, −6.17733322398346711377549901779, −5.05560859742995612622852506480, −4.09125756483153066134175366350, −2.88879412407809924508597793019, −2.27891319545199188336160611014, 1.60602509462789368011036820447, 3.45284677515768710478987857766, 4.19008088941263594266512835086, 5.27315745862129968827779991655, 6.53269485417910468744953759329, 7.38769155056640560723257076458, 8.061100101424918135426368680152, 9.381775398215754935433556554315, 10.11101818462718024870098935928, 10.97382435243125854241210030448

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