Properties

Label 2-21e2-21.20-c3-0-14
Degree $2$
Conductor $441$
Sign $0.970 + 0.239i$
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  − 1.82i·2-s + 4.68·4-s − 15.0·5-s − 23.0i·8-s + 27.4i·10-s + 9.89i·11-s + 67.8i·13-s − 4.54·16-s + 70.1·17-s + 61.4i·19-s − 70.7·20-s + 18.0·22-s − 131. i·23-s + 102.·25-s + 123.·26-s + ⋯
L(s)  = 1  − 0.643i·2-s + 0.585·4-s − 1.34·5-s − 1.02i·8-s + 0.868i·10-s + 0.271i·11-s + 1.44i·13-s − 0.0710·16-s + 1.00·17-s + 0.742i·19-s − 0.790·20-s + 0.174·22-s − 1.19i·23-s + 0.821·25-s + 0.932·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.970 + 0.239i$
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: \((2,\ 441,\ (\ :3/2),\ 0.970 + 0.239i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.739260588\)
\(L(\frac12)\) \(\approx\) \(1.739260588\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.82iT - 8T^{2} \)
5 \( 1 + 15.0T + 125T^{2} \)
11 \( 1 - 9.89iT - 1.33e3T^{2} \)
13 \( 1 - 67.8iT - 2.19e3T^{2} \)
17 \( 1 - 70.1T + 4.91e3T^{2} \)
19 \( 1 - 61.4iT - 6.85e3T^{2} \)
23 \( 1 + 131. iT - 1.21e4T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 + 76.4iT - 2.97e4T^{2} \)
37 \( 1 - 348.T + 5.06e4T^{2} \)
41 \( 1 + 138.T + 6.89e4T^{2} \)
43 \( 1 - 539.T + 7.95e4T^{2} \)
47 \( 1 - 223.T + 1.03e5T^{2} \)
53 \( 1 - 530. iT - 1.48e5T^{2} \)
59 \( 1 - 542.T + 2.05e5T^{2} \)
61 \( 1 - 134. iT - 2.26e5T^{2} \)
67 \( 1 - 320.T + 3.00e5T^{2} \)
71 \( 1 + 416. iT - 3.57e5T^{2} \)
73 \( 1 - 545. iT - 3.89e5T^{2} \)
79 \( 1 + 322.T + 4.93e5T^{2} \)
83 \( 1 + 885.T + 5.71e5T^{2} \)
89 \( 1 - 1.62e3T + 7.04e5T^{2} \)
97 \( 1 + 739. iT - 9.12e5T^{2} \)
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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

−10.88786861473611051611520798819, −9.989364061384210179964632427537, −8.926821899379578071867905227792, −7.76408187373839826792904976656, −7.14359388496003971060654543031, −6.08454570215618172082961422929, −4.39859412438692122421031825411, −3.71570286506940199834339276914, −2.45262471232157360928187675231, −1.01149850023005358664082557902, 0.71849247831558934727287399603, 2.75226741232278185299899707067, 3.75541897116314389120448540590, 5.19306180218415961152716827817, 6.03859816855554030566109032946, 7.36199610333523379548817174242, 7.74680516935687237518336428667, 8.488926408543118205116238303403, 9.890999103652848538526418334611, 10.98411120346213101374696047814

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