Properties

Degree $2$
Conductor $441$
Sign $0.992 + 0.123i$
Motivic weight $1$
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: $0.992 + 0.123i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(2.02010 - 0.124721i\)
\(L(\frac12)\) \(\approx\) \(2.02010 - 0.124721i\)
\(L(\frac{3}{2})\) 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

−11.14435378428317849960424621701, −10.37841780159068823234141306453, −8.717248851647134687593769244997, −8.185407239177601433819952141790, −7.23240405275684026615319940796, −6.28737061839356618960427922584, −5.64436523997640977632602619900, −4.77634349254236418595201062956, −3.06836368120998322368924491319, −1.22260651346530075232543579688, 1.96792868100166654967151973111, 3.21157133010579593866687050714, 4.06272244480376226873089402571, 4.98832884238588984666438495865, 6.13431219933473369863710333264, 7.39239411204390376801812626092, 8.844306265854680697487566256969, 9.689657081309061728616071790958, 10.51536196067358687777987793922, 11.11819384552543062065649964042

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