Properties

Label 2-768-4.3-c6-0-77
Degree $2$
Conductor $768$
Sign $i$
Analytic cond. $176.681$
Root an. cond. $13.2921$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 15.5i·3-s + 20·5-s + 529. i·7-s − 243·9-s − 435. i·11-s + 341.·13-s + 311. i·15-s + 7.68e3·17-s − 4.30e3i·19-s − 8.25e3·21-s − 3.17e3i·23-s − 1.52e4·25-s − 3.78e3i·27-s − 1.94e4·29-s − 1.52e4i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.160·5-s + 1.54i·7-s − 0.333·9-s − 0.327i·11-s + 0.155·13-s + 0.0923i·15-s + 1.56·17-s − 0.626i·19-s − 0.891·21-s − 0.260i·23-s − 0.974·25-s − 0.192i·27-s − 0.795·29-s − 0.513i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $i$
Analytic conductor: \(176.681\)
Root analytic conductor: \(13.2921\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3),\ i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4745247520\)
\(L(\frac12)\) \(\approx\) \(0.4745247520\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 15.5iT \)
good5 \( 1 - 20T + 1.56e4T^{2} \)
7 \( 1 - 529. iT - 1.17e5T^{2} \)
11 \( 1 + 435. iT - 1.77e6T^{2} \)
13 \( 1 - 341.T + 4.82e6T^{2} \)
17 \( 1 - 7.68e3T + 2.41e7T^{2} \)
19 \( 1 + 4.30e3iT - 4.70e7T^{2} \)
23 \( 1 + 3.17e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.94e4T + 5.94e8T^{2} \)
31 \( 1 + 1.52e4iT - 8.87e8T^{2} \)
37 \( 1 + 6.19e4T + 2.56e9T^{2} \)
41 \( 1 - 3.37e4T + 4.75e9T^{2} \)
43 \( 1 - 9.93e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.77e4iT - 1.07e10T^{2} \)
53 \( 1 + 2.24e5T + 2.21e10T^{2} \)
59 \( 1 + 1.99e5iT - 4.21e10T^{2} \)
61 \( 1 + 4.56e4T + 5.15e10T^{2} \)
67 \( 1 + 4.96e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.52e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.94e5T + 1.51e11T^{2} \)
79 \( 1 + 5.71e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.24e5iT - 3.26e11T^{2} \)
89 \( 1 + 7.58e5T + 4.96e11T^{2} \)
97 \( 1 - 2.50e4T + 8.32e11T^{2} \)
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

−9.322550888255314622618754362969, −8.444292635689566685650756943902, −7.66885969381807754609573941470, −6.22037973511594729895242222637, −5.65253236282025085043465111414, −4.86890577807709829379306453980, −3.55008650128751391938943071632, −2.73912962020670114910676725592, −1.63373687944184562036791185365, −0.089950355601119815122092482150, 1.05174078842176385310008089776, 1.77211672189695461379539933619, 3.31286786846642098069004317777, 4.01865744487114155387028547725, 5.29132945798812208399986519856, 6.16680140570973856315486226531, 7.41626990842515354606735648984, 7.46347811403662293026094272913, 8.628108484342114762272315812287, 9.864616870413459492685337487957

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