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Properties

Label 2-2e8-4.3-c10-0-28
Degree $2$
Conductor $256$
Sign $-1$
Analytic cond. $162.651$
Root an. cond. $12.7534$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 242. i·3-s − 3.14e3·5-s + 1.86e4i·7-s + 9·9-s + 8.50e3i·11-s + 2.11e5·13-s − 7.65e5i·15-s − 4.35e3·17-s − 8.56e5i·19-s − 4.53e6·21-s + 5.50e6i·23-s + 1.53e5·25-s + 1.43e7i·27-s + 2.70e7·29-s + 4.91e7i·31-s + ⋯
L(s)  = 1  + 0.999i·3-s − 1.00·5-s + 1.11i·7-s + 0.000152·9-s + 0.0528i·11-s + 0.568·13-s − 1.00i·15-s − 0.00306·17-s − 0.345i·19-s − 1.11·21-s + 0.855i·23-s + 0.0156·25-s + 1.00i·27-s + 1.32·29-s + 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-1$
Analytic conductor: \(162.651\)
Root analytic conductor: \(12.7534\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5),\ -1)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.750361380\)
\(L(\frac12)\) \(\approx\) \(1.750361380\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 242. iT - 5.90e4T^{2} \)
5 \( 1 + 3.14e3T + 9.76e6T^{2} \)
7 \( 1 - 1.86e4iT - 2.82e8T^{2} \)
11 \( 1 - 8.50e3iT - 2.59e10T^{2} \)
13 \( 1 - 2.11e5T + 1.37e11T^{2} \)
17 \( 1 + 4.35e3T + 2.01e12T^{2} \)
19 \( 1 + 8.56e5iT - 6.13e12T^{2} \)
23 \( 1 - 5.50e6iT - 4.14e13T^{2} \)
29 \( 1 - 2.70e7T + 4.20e14T^{2} \)
31 \( 1 - 4.91e7iT - 8.19e14T^{2} \)
37 \( 1 - 8.89e7T + 4.80e15T^{2} \)
41 \( 1 - 1.30e7T + 1.34e16T^{2} \)
43 \( 1 - 2.31e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.35e8iT - 5.25e16T^{2} \)
53 \( 1 - 4.40e8T + 1.74e17T^{2} \)
59 \( 1 - 3.73e8iT - 5.11e17T^{2} \)
61 \( 1 - 8.17e8T + 7.13e17T^{2} \)
67 \( 1 - 2.34e9iT - 1.82e18T^{2} \)
71 \( 1 + 2.21e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.40e9T + 4.29e18T^{2} \)
79 \( 1 + 1.36e9iT - 9.46e18T^{2} \)
83 \( 1 - 6.04e9iT - 1.55e19T^{2} \)
89 \( 1 - 5.47e9T + 3.11e19T^{2} \)
97 \( 1 + 1.18e10T + 7.37e19T^{2} \)
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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

−10.74299647613322986499205893359, −9.721473269249077206337446113986, −8.854593415939304728344256832821, −8.063833361811522957376632043911, −6.82337710121763881282481890930, −5.54869808428023973344785412299, −4.60398538587982885523293839351, −3.70923890634046436091231714106, −2.73235433432372180320808352502, −1.13907985647333789445234457480, 0.45436944620183120163293139201, 0.940830734157614037787113102417, 2.29660068903977319068796951765, 3.76592651402436670685561749691, 4.41252245836326146182852487341, 6.11342203470641310728176983721, 7.00994552714451730557184582941, 7.74842924248097214792497416277, 8.384571829824331007947707439996, 9.895539625112910429167259972652

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