Properties

Label 2-21e2-9.7-c1-0-32
Degree $2$
Conductor $441$
Sign $-0.992 + 0.123i$
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  + (0.920 − 1.59i)2-s + (0.195 − 1.72i)3-s + (−0.695 − 1.20i)4-s + (−0.667 − 1.15i)5-s + (−2.56 − 1.89i)6-s + 1.12·8-s + (−2.92 − 0.671i)9-s − 2.45·10-s + (−0.756 + 1.31i)11-s + (−2.20 + 0.961i)12-s + (−2.58 − 4.48i)13-s + (−2.11 + 0.923i)15-s + (2.42 − 4.19i)16-s − 1.54·17-s + (−3.76 + 4.04i)18-s + 2.50·19-s + ⋯
L(s)  = 1  + (0.650 − 1.12i)2-s + (0.112 − 0.993i)3-s + (−0.347 − 0.601i)4-s + (−0.298 − 0.516i)5-s + (−1.04 − 0.773i)6-s + 0.396·8-s + (−0.974 − 0.223i)9-s − 0.777·10-s + (−0.228 + 0.395i)11-s + (−0.637 + 0.277i)12-s + (−0.717 − 1.24i)13-s + (−0.547 + 0.238i)15-s + (0.605 − 1.04i)16-s − 0.375·17-s + (−0.886 + 0.953i)18-s + 0.574·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.992 + 0.123i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113665 - 1.84102i\)
\(L(\frac12)\) \(\approx\) \(0.113665 - 1.84102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.195 + 1.72i)T \)
7 \( 1 \)
good2 \( 1 + (-0.920 + 1.59i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.667 + 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.756 - 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.54T + 17T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 + (-3.68 - 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0309 - 0.0536i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.563T + 37T^{2} \)
41 \( 1 + (-4.51 - 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.75 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.51T + 53T^{2} \)
59 \( 1 + (-4.22 - 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 2.75T + 73T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.80 + 4.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.40T + 89T^{2} \)
97 \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.97203140076499682754816532441, −10.07835286844338789575714157421, −8.901214937243127161491535267416, −7.76910701344671053456619813378, −7.21733048900266389350255670338, −5.61093291541355494956422285969, −4.76014264261970350523234892545, −3.33645980256861515037662095818, −2.41618374184675178991588137517, −0.992603238637010132077697014403, 2.73897780733501486760334799121, 4.12765972186074958473395302462, 4.78853633588598868935177685285, 5.85678832941456856604486291738, 6.81196063864009735304361143600, 7.65965748903180179033362608029, 8.763224187935964324907055501528, 9.654126442479542074553664217103, 10.79924575233646730019983875610, 11.28094886560431646174491387926

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