Properties

Label 2-21e2-441.122-c1-0-22
Degree $2$
Conductor $441$
Sign $0.857 - 0.514i$
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  + (−1.22 + 1.32i)2-s + (0.168 − 1.72i)3-s + (−0.0944 − 1.25i)4-s + (0.684 − 0.858i)5-s + (2.07 + 2.34i)6-s + (1.06 + 2.42i)7-s + (−1.04 − 0.829i)8-s + (−2.94 − 0.582i)9-s + (0.295 + 1.96i)10-s + (4.90 + 1.11i)11-s + (−2.18 − 0.0500i)12-s + (−2.31 + 2.49i)13-s + (−4.51 − 1.56i)14-s + (−1.36 − 1.32i)15-s + (4.87 − 0.734i)16-s + (0.0541 − 0.722i)17-s + ⋯
L(s)  = 1  + (−0.868 + 0.936i)2-s + (0.0975 − 0.995i)3-s + (−0.0472 − 0.629i)4-s + (0.306 − 0.383i)5-s + (0.847 + 0.955i)6-s + (0.402 + 0.915i)7-s + (−0.367 − 0.293i)8-s + (−0.980 − 0.194i)9-s + (0.0934 + 0.620i)10-s + (1.47 + 0.337i)11-s + (−0.631 − 0.0144i)12-s + (−0.642 + 0.692i)13-s + (−1.20 − 0.418i)14-s + (−0.352 − 0.342i)15-s + (1.21 − 0.183i)16-s + (0.0131 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.981468 + 0.272100i\)
\(L(\frac12)\) \(\approx\) \(0.981468 + 0.272100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.168 + 1.72i)T \)
7 \( 1 + (-1.06 - 2.42i)T \)
good2 \( 1 + (1.22 - 1.32i)T + (-0.149 - 1.99i)T^{2} \)
5 \( 1 + (-0.684 + 0.858i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-4.90 - 1.11i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (2.31 - 2.49i)T + (-0.971 - 12.9i)T^{2} \)
17 \( 1 + (-0.0541 + 0.722i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-3.19 + 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.87 + 3.89i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-4.96 + 7.27i)T + (-10.5 - 26.9i)T^{2} \)
31 \( 1 + (-5.05 + 2.91i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.62 - 5.87i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (0.00356 + 0.00907i)T + (-30.0 + 27.8i)T^{2} \)
43 \( 1 + (1.60 - 4.09i)T + (-31.5 - 29.2i)T^{2} \)
47 \( 1 + (-0.570 - 0.529i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (0.0965 + 0.141i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-1.11 + 2.83i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (13.4 + 1.01i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-2.96 - 5.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.66 - 9.69i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-3.82 + 12.4i)T + (-60.3 - 41.1i)T^{2} \)
79 \( 1 + (-3.99 + 6.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.60 - 7.06i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (-5.96 + 5.53i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-11.6 + 6.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.63903160448105096471107326145, −9.624677881568922779248245313426, −9.211087938310044655053090687040, −8.408798501208371895679602159317, −7.58425077209867553187475767116, −6.60430312087512599362519884516, −6.09163015895495122508113514490, −4.71358112991271357957291628753, −2.70067077743719190696858627615, −1.19970168846098123115111739315, 1.14327541901875752699771862031, 2.85578620147535095737336855183, 3.80549243509367632003211372501, 5.10562776742331600955414423393, 6.33238709700768687914929403868, 7.71613335250958801159693006241, 8.738793115591788153341466743893, 9.505248335220964366817278868081, 10.23367325728749030203460864364, 10.74285595117595526691455502106

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