↑

Properties

Label 2-768-24.11-c1-0-4
Degree $2$
Conductor $768$
Sign $-0.912 - 0.408i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 + 1.61i)3-s − 1.23·5-s + 3.23i·7-s + (−2.23 + 2.00i)9-s + 0.763i·11-s − 4.47i·13-s + (−0.763 − 2.00i)15-s + 6.47i·17-s + 5.23·19-s + (−5.23 + 2.00i)21-s − 6.47·23-s − 3.47·25-s + (−4.61 − 2.38i)27-s − 9.23·29-s + 0.763i·31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s − 0.552·5-s + 1.22i·7-s + (−0.745 + 0.666i)9-s + 0.230i·11-s − 1.24i·13-s + (−0.197 − 0.516i)15-s + 1.56i·17-s + 1.20·19-s + (−1.14 + 0.436i)21-s − 1.34·23-s − 0.694·25-s + (−0.888 − 0.458i)27-s − 1.71·29-s + 0.137i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227160 + 1.06437i\)
\(L(\frac12)\) \(\approx\) \(0.227160 + 1.06437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 - 1.61i)T \)
good5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 3.23iT - 7T^{2} \)
11 \( 1 - 0.763iT - 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 - 6.47iT - 17T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + 9.23T + 29T^{2} \)
31 \( 1 - 0.763iT - 31T^{2} \)
37 \( 1 + 0.472iT - 37T^{2} \)
41 \( 1 - 2.47iT - 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 1.23T + 53T^{2} \)
59 \( 1 - 3.23iT - 59T^{2} \)
61 \( 1 - 8.47iT - 61T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 + 7.23iT - 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 8.47T + 97T^{2} \)
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

−10.56534956959607588420773050672, −9.821383611616386129698740212209, −9.019400553888908659716907101488, −8.153483741653048215105475653549, −7.64989126945510877447843711283, −5.80571108087965117853067408926, −5.54760393601450449507557420544, −4.13779310981442402282961542186, −3.36482183168630725341692540379, −2.17739300799737743999073316964, 0.51125426464725027367664674257, 1.96498566430316388864557059423, 3.42331505601323547909248293365, 4.21776008711077675115181548693, 5.60956950257273861125774565844, 6.78943000404672201453841396795, 7.44247778666628814458508171474, 7.86835287838664711092733213857, 9.167665380491056482681912782638, 9.716870417847522983425860210478

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