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Properties

Label 2-21e2-9.4-c1-0-7
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: $\chi_{441} (148, \cdot )$
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

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

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