Properties

Label 2-21e2-21.20-c3-0-12
Degree $2$
Conductor $441$
Sign $-0.192 - 0.981i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.38i·2-s − 20.9·4-s − 18.3·5-s − 69.8i·8-s − 98.6i·10-s − 23.7i·11-s + 11.7i·13-s + 208.·16-s − 96.4·17-s − 21.0i·19-s + 384.·20-s + 127.·22-s + 152. i·23-s + 210.·25-s − 63.5·26-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.62·4-s − 1.63·5-s − 3.08i·8-s − 3.11i·10-s − 0.651i·11-s + 0.251i·13-s + 3.25·16-s − 1.37·17-s − 0.253i·19-s + 4.29·20-s + 1.23·22-s + 1.38i·23-s + 1.68·25-s − 0.478·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6209521678\)
\(L(\frac12)\) \(\approx\) \(0.6209521678\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 5.38iT - 8T^{2} \)
5 \( 1 + 18.3T + 125T^{2} \)
11 \( 1 + 23.7iT - 1.33e3T^{2} \)
13 \( 1 - 11.7iT - 2.19e3T^{2} \)
17 \( 1 + 96.4T + 4.91e3T^{2} \)
19 \( 1 + 21.0iT - 6.85e3T^{2} \)
23 \( 1 - 152. iT - 1.21e4T^{2} \)
29 \( 1 - 77.5iT - 2.43e4T^{2} \)
31 \( 1 + 199. iT - 2.97e4T^{2} \)
37 \( 1 - 164.T + 5.06e4T^{2} \)
41 \( 1 + 435.T + 6.89e4T^{2} \)
43 \( 1 + 106.T + 7.95e4T^{2} \)
47 \( 1 - 11.0T + 1.03e5T^{2} \)
53 \( 1 - 110. iT - 1.48e5T^{2} \)
59 \( 1 - 829.T + 2.05e5T^{2} \)
61 \( 1 + 22.5iT - 2.26e5T^{2} \)
67 \( 1 - 398.T + 3.00e5T^{2} \)
71 \( 1 + 228. iT - 3.57e5T^{2} \)
73 \( 1 - 266. iT - 3.89e5T^{2} \)
79 \( 1 - 920.T + 4.93e5T^{2} \)
83 \( 1 + 642.T + 5.71e5T^{2} \)
89 \( 1 - 445.T + 7.04e5T^{2} \)
97 \( 1 - 1.61e3iT - 9.12e5T^{2} \)
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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

−11.06855049757602210795749817024, −9.553922438563175739551247535308, −8.643152700106831687676612859912, −8.082297668281181785282024511703, −7.23771537747491060502972885374, −6.56526632051916678764608664046, −5.35109790678227115248149633027, −4.35875705373978462314574254062, −3.58533521044323061200118233737, −0.41094310701471828200919706622, 0.58984062420914842167840678731, 2.19354757854843679936273371216, 3.35896982062368105653203022524, 4.25874228216624919670704777010, 4.87753746869044404220140051652, 6.89774518080800707195040083069, 8.240307803014562175495569854380, 8.684250702566612185401629379288, 9.902208555806179851801325987929, 10.70630151995561929444023495192

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