Properties

Label 2-2e8-8.3-c8-0-20
Degree $2$
Conductor $256$
Sign $0.707 - 0.707i$
Analytic cond. $104.288$
Root an. cond. $10.2121$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 99.9·3-s + 610i·5-s + 1.39e3i·7-s + 3.42e3·9-s − 1.84e4·11-s + 5.47e3i·13-s − 6.09e4i·15-s + 7.30e4·17-s + 1.94e4·19-s − 1.39e5i·21-s − 2.37e5i·23-s + 1.85e4·25-s + 3.13e5·27-s + 1.28e5i·29-s + 6.79e4i·31-s + ⋯
L(s)  = 1  − 1.23·3-s + 0.976i·5-s + 0.582i·7-s + 0.521·9-s − 1.26·11-s + 0.191i·13-s − 1.20i·15-s + 0.875·17-s + 0.149·19-s − 0.718i·21-s − 0.847i·23-s + 0.0474·25-s + 0.589·27-s + 0.181i·29-s + 0.0735i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(104.288\)
Root analytic conductor: \(10.2121\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :4),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.8773902364\)
\(L(\frac12)\) \(\approx\) \(0.8773902364\)
\(L(5)\) 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 \( 1 + 99.9T + 6.56e3T^{2} \)
5 \( 1 - 610iT - 3.90e5T^{2} \)
7 \( 1 - 1.39e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.84e4T + 2.14e8T^{2} \)
13 \( 1 - 5.47e3iT - 8.15e8T^{2} \)
17 \( 1 - 7.30e4T + 6.97e9T^{2} \)
19 \( 1 - 1.94e4T + 1.69e10T^{2} \)
23 \( 1 + 2.37e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.28e5iT - 5.00e11T^{2} \)
31 \( 1 - 6.79e4iT - 8.52e11T^{2} \)
37 \( 1 + 3.47e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.14e6T + 7.98e12T^{2} \)
43 \( 1 + 5.92e6T + 1.16e13T^{2} \)
47 \( 1 + 7.62e6iT - 2.38e13T^{2} \)
53 \( 1 - 8.24e5iT - 6.22e13T^{2} \)
59 \( 1 + 3.72e6T + 1.46e14T^{2} \)
61 \( 1 - 1.47e7iT - 1.91e14T^{2} \)
67 \( 1 + 1.52e7T + 4.06e14T^{2} \)
71 \( 1 + 1.19e6iT - 6.45e14T^{2} \)
73 \( 1 - 5.72e6T + 8.06e14T^{2} \)
79 \( 1 + 3.59e7iT - 1.51e15T^{2} \)
83 \( 1 - 5.19e7T + 2.25e15T^{2} \)
89 \( 1 - 8.33e7T + 3.93e15T^{2} \)
97 \( 1 - 1.20e8T + 7.83e15T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.55365059302029412355707488305, −10.32667366396890183363638952350, −8.776853810765166589830615027498, −7.55199525783005403177197091128, −6.60246988791628375479050010079, −5.67295804813667664672790733388, −4.96861525798825246603678784058, −3.29421851909459324524517000417, −2.21768741122885016240363351138, −0.49670942186056966572309288554, 0.48517083897071666792839321961, 1.37007022894177337222404099398, 3.20494915335211756313711372669, 4.79914844610955696274167345929, 5.22961083214542580409781357652, 6.27462971809906186652444149985, 7.55048903423885961416800616489, 8.401765298321495865247861992403, 9.802423280759451623917092007922, 10.49990227843135655180263861792

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