Properties

Label 2-2e8-8.3-c6-0-0
Degree $2$
Conductor $256$
Sign $0.707 - 0.707i$
Analytic cond. $58.8938$
Root an. cond. $7.67423$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27.1·3-s − 170. i·5-s − 374. i·7-s + 10.3·9-s − 1.35e3·11-s + 2.95e3i·13-s + 4.64e3i·15-s − 6.65e3·17-s − 9.92e3·19-s + 1.01e4i·21-s − 1.55e4i·23-s − 1.35e4·25-s + 1.95e4·27-s + 2.08e3i·29-s − 2.84e4i·31-s + ⋯
L(s)  = 1  − 1.00·3-s − 1.36i·5-s − 1.09i·7-s + 0.0142·9-s − 1.02·11-s + 1.34i·13-s + 1.37i·15-s − 1.35·17-s − 1.44·19-s + 1.09i·21-s − 1.28i·23-s − 0.866·25-s + 0.992·27-s + 0.0854i·29-s − 0.954i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(58.8938\)
Root analytic conductor: \(7.67423\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.1419175768\)
\(L(\frac12)\) \(\approx\) \(0.1419175768\)
\(L(4)\) 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 \( 1 + 27.1T + 729T^{2} \)
5 \( 1 + 170. iT - 1.56e4T^{2} \)
7 \( 1 + 374. iT - 1.17e5T^{2} \)
11 \( 1 + 1.35e3T + 1.77e6T^{2} \)
13 \( 1 - 2.95e3iT - 4.82e6T^{2} \)
17 \( 1 + 6.65e3T + 2.41e7T^{2} \)
19 \( 1 + 9.92e3T + 4.70e7T^{2} \)
23 \( 1 + 1.55e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.08e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.84e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.33e3iT - 2.56e9T^{2} \)
41 \( 1 - 6.31e4T + 4.75e9T^{2} \)
43 \( 1 + 8.85e4T + 6.32e9T^{2} \)
47 \( 1 + 1.08e4iT - 1.07e10T^{2} \)
53 \( 1 + 4.80e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.47e5T + 4.21e10T^{2} \)
61 \( 1 - 1.24e5iT - 5.15e10T^{2} \)
67 \( 1 + 8.75e4T + 9.04e10T^{2} \)
71 \( 1 - 2.25e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.75e5T + 1.51e11T^{2} \)
79 \( 1 + 2.76e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.85e5T + 3.26e11T^{2} \)
89 \( 1 - 1.06e6T + 4.96e11T^{2} \)
97 \( 1 - 1.46e6T + 8.32e11T^{2} \)
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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

−11.03760418927635952692164200648, −10.40775968483372750770370440194, −9.049140938033573882022703097391, −8.303956510330706942314760872674, −6.92585254266002714279446623709, −6.03117875142580621676197384945, −4.61571718551264554816129019840, −4.42453819480886603720108426043, −2.09058915763488323716753325477, −0.60605968163710023699374893698, 0.06819147467122113175911184348, 2.28342529015102557418217955744, 3.13029759278365549684827570358, 4.97715889920023008265343124981, 5.87848242140938450208885948890, 6.56669526715099703348915804554, 7.79680181886021966390151586050, 8.887483866130564691324783009323, 10.41518957990534671870331458602, 10.76407113379890573336994563440

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