Properties

Label 2-21e2-21.20-c3-0-15
Degree $2$
Conductor $441$
Sign $-0.0980 - 0.995i$
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  + 4.86i·2-s − 15.6·4-s − 12.7·5-s − 37.3i·8-s − 62.1i·10-s − 54.1i·11-s − 8.85i·13-s + 56.2·16-s + 68.9·17-s + 163. i·19-s + 200.·20-s + 263.·22-s − 93.9i·23-s + 37.9·25-s + 43.0·26-s + ⋯
L(s)  = 1  + 1.72i·2-s − 1.95·4-s − 1.14·5-s − 1.64i·8-s − 1.96i·10-s − 1.48i·11-s − 0.188i·13-s + 0.878·16-s + 0.983·17-s + 1.97i·19-s + 2.23·20-s + 2.55·22-s − 0.851i·23-s + 0.303·25-s + 0.324·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0980 - 0.995i$
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: \((2,\ 441,\ (\ :3/2),\ -0.0980 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.114579802\)
\(L(\frac12)\) \(\approx\) \(1.114579802\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.86iT - 8T^{2} \)
5 \( 1 + 12.7T + 125T^{2} \)
11 \( 1 + 54.1iT - 1.33e3T^{2} \)
13 \( 1 + 8.85iT - 2.19e3T^{2} \)
17 \( 1 - 68.9T + 4.91e3T^{2} \)
19 \( 1 - 163. iT - 6.85e3T^{2} \)
23 \( 1 + 93.9iT - 1.21e4T^{2} \)
29 \( 1 + 119. iT - 2.43e4T^{2} \)
31 \( 1 - 98.8iT - 2.97e4T^{2} \)
37 \( 1 - 94.1T + 5.06e4T^{2} \)
41 \( 1 - 259.T + 6.89e4T^{2} \)
43 \( 1 - 5.01T + 7.95e4T^{2} \)
47 \( 1 + 57.3T + 1.03e5T^{2} \)
53 \( 1 - 470. iT - 1.48e5T^{2} \)
59 \( 1 - 225.T + 2.05e5T^{2} \)
61 \( 1 + 427. iT - 2.26e5T^{2} \)
67 \( 1 - 163.T + 3.00e5T^{2} \)
71 \( 1 + 79.8iT - 3.57e5T^{2} \)
73 \( 1 - 769. iT - 3.89e5T^{2} \)
79 \( 1 - 534.T + 4.93e5T^{2} \)
83 \( 1 - 438.T + 5.71e5T^{2} \)
89 \( 1 + 25.6T + 7.04e5T^{2} \)
97 \( 1 + 1.38e3iT - 9.12e5T^{2} \)
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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

−10.91324911521582694163040753505, −9.766119825317747454018489323976, −8.481585550364094438085493114006, −8.111011303489316117273833126073, −7.42867145227371881667640538578, −6.17087551840074081569534425244, −5.62779577992597780008613305705, −4.29240059027393556410520260538, −3.40331653230496979546404011331, −0.60362025812336524623301746553, 0.75182704675420247217558092807, 2.17061358834355910707038972717, 3.32886645065176058336533053958, 4.27188497622941433070188493108, 5.04206583619785448342283122815, 7.01314521432208620747918887033, 7.84242692011508809869052621292, 9.108681528805160981960803305199, 9.695267157672675149633665253066, 10.68487891882800203542553623200

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