↑

Properties

Label 2-768-24.11-c1-0-0
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

\(L(1)\) \(\approx\) \(0.0921803 - 0.431916i\)
\(L(\frac12)\) \(\approx\) \(0.0921803 - 0.431916i\)
\(L(\frac{3}{2})\) 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

−11.05930996089870945074026964490, −9.738315316494419976777170980624, −9.178469164861366366257637682500, −8.585964302788056660269182102667, −7.27426726825701418649957522303, −6.33311189130423687249434381240, −5.28054154575686224755064032135, −4.58007398019946576177703920396, −3.50423865323254091183318992841, −2.27226121625915783222418550247, 0.22898443236149848169555542180, 1.61921106160247963726547859665, 3.26251381525882342521707204672, 4.24888606900431887658516334376, 5.53713211455925263401734154656, 6.40942945489643600095796549278, 7.34601987751179902385258766770, 7.937518760407623747524673603422, 8.652782704242151537695459857454, 10.14172047491469205434816040678

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