Properties

Label 2-21e2-63.16-c1-0-33
Degree $2$
Conductor $441$
Sign $-0.953 - 0.301i$
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  + (−0.649 − 1.12i)2-s + (1.52 − 0.821i)3-s + (0.155 − 0.268i)4-s − 3.52·5-s + (−1.91 − 1.18i)6-s − 3.00·8-s + (1.65 − 2.50i)9-s + (2.29 + 3.96i)10-s + 1.17·11-s + (0.0159 − 0.537i)12-s + (−1.61 − 2.78i)13-s + (−5.37 + 2.89i)15-s + (1.64 + 2.84i)16-s + (−2.45 − 4.24i)17-s + (−3.89 − 0.231i)18-s + (−3.43 + 5.94i)19-s + ⋯
L(s)  = 1  + (−0.459 − 0.796i)2-s + (0.880 − 0.474i)3-s + (0.0775 − 0.134i)4-s − 1.57·5-s + (−0.782 − 0.482i)6-s − 1.06·8-s + (0.550 − 0.834i)9-s + (0.724 + 1.25i)10-s + 0.355·11-s + (0.00460 − 0.155i)12-s + (−0.446 − 0.773i)13-s + (−1.38 + 0.747i)15-s + (0.410 + 0.710i)16-s + (−0.594 − 1.02i)17-s + (−0.917 − 0.0545i)18-s + (−0.787 + 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.953 - 0.301i$
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: \((2,\ 441,\ (\ :1/2),\ -0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117512 + 0.760724i\)
\(L(\frac12)\) \(\approx\) \(0.117512 + 0.760724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.52 + 0.821i)T \)
7 \( 1 \)
good2 \( 1 + (0.649 + 1.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.52T + 5T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + (1.61 + 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.45 + 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.43 - 5.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.29T + 23T^{2} \)
29 \( 1 + (-1.36 + 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.960 - 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.88 + 8.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.32 - 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.316 - 0.548i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.11 - 1.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.10 + 7.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.82 + 8.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.66 - 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + (-0.519 - 0.898i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.502 + 0.869i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.65 + 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.02 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.46 + 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66205668351681094844991320222, −9.727737208972131735412374902558, −8.806995460180610280480131612358, −7.995944655905246046141411132845, −7.28677715829369150521429748038, −6.07822074686832329119031700078, −4.25557553066317885291572761896, −3.33771978166774825449346868352, −2.19558972044110696116252551893, −0.48656251661312492981554787381, 2.60713387764308263757441329308, 3.88777203420201498600217023845, 4.51750989948506456143329431282, 6.44350296313580587434836060720, 7.28319905019939679665480785131, 8.028167757419751888243441715225, 8.681065230353359329566804557025, 9.346972447276776322308542801420, 10.71625342246816336442201399187, 11.56917871628994559114775579909

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