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Properties

Label 2-21e2-3.2-c0-0-1
Degree $2$
Conductor $441$
Sign $-0.577 + 0.816i$
Analytic cond. $0.220087$
Root an. cond. $0.469135$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 1.41i·11-s − 0.999·16-s − 2.00·22-s + 1.41i·23-s + 25-s + 1.41i·29-s + 1.41i·32-s + 1.41i·44-s + 2.00·46-s − 1.41i·50-s + 1.41i·53-s + 2.00·58-s + 1.00·64-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 1.41i·11-s − 0.999·16-s − 2.00·22-s + 1.41i·23-s + 25-s + 1.41i·29-s + 1.41i·32-s + 1.41i·44-s + 2.00·46-s − 1.41i·50-s + 1.41i·53-s + 2.00·58-s + 1.00·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8415097293\)
\(L(\frac12)\) \(\approx\) \(0.8415097293\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 1.41iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.96857346074114353539039441239, −10.51651999613889885513362405791, −9.329485489699679464914913646392, −8.737212900068718365040170104805, −7.44273120123202670099339716443, −6.21468501338760857365001128365, −5.01233410038919233840908818842, −3.64577065757865264279999126722, −2.91671596863141148818752592879, −1.32837335488096312434276989129, 2.36438985690773399053883083814, 4.31225387507632891502908836307, 5.07896495965005685939838322715, 6.28449274811526203539564201486, 6.96600112572225974598710138211, 7.82066734516852205356545624700, 8.669173095535633182753122905473, 9.637006426297288684903192565796, 10.58591257591691448054745128864, 11.75768382820734119807401977288

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