↑

Properties

Label 2-768-24.11-c3-0-72
Degree $2$
Conductor $768$
Sign $-0.0556 + 0.998i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.87 − 3.46i)3-s + 8.94·5-s + 7.74i·7-s + (3.00 − 26.8i)9-s − 34.6i·11-s + 10i·13-s + (34.6 − 30.9i)15-s − 35.7i·17-s + 69.7·19-s + (26.8 + 30.0i)21-s − 96.9·23-s − 44.9·25-s + (−81.3 − 114. i)27-s + 152.·29-s − 224. i·31-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)3-s + 0.799·5-s + 0.418i·7-s + (0.111 − 0.993i)9-s − 0.949i·11-s + 0.213i·13-s + (0.596 − 0.533i)15-s − 0.510i·17-s + 0.841·19-s + (0.278 + 0.311i)21-s − 0.879·23-s − 0.359·25-s + (−0.579 − 0.814i)27-s + 0.973·29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0556 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0556 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.0556 + 0.998i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.0556 + 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.944714479\)
\(L(\frac12)\) \(\approx\) \(2.944714479\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.87 + 3.46i)T \)
good5 \( 1 - 8.94T + 125T^{2} \)
7 \( 1 - 7.74iT - 343T^{2} \)
11 \( 1 + 34.6iT - 1.33e3T^{2} \)
13 \( 1 - 10iT - 2.19e3T^{2} \)
17 \( 1 + 35.7iT - 4.91e3T^{2} \)
19 \( 1 - 69.7T + 6.85e3T^{2} \)
23 \( 1 + 96.9T + 1.21e4T^{2} \)
29 \( 1 - 152.T + 2.43e4T^{2} \)
31 \( 1 + 224. iT - 2.97e4T^{2} \)
37 \( 1 + 130iT - 5.06e4T^{2} \)
41 \( 1 + 125. iT - 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 - 193.T + 1.03e5T^{2} \)
53 \( 1 - 545.T + 1.48e5T^{2} \)
59 \( 1 + 173. iT - 2.05e5T^{2} \)
61 \( 1 - 442iT - 2.26e5T^{2} \)
67 \( 1 + 735.T + 3.00e5T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 410T + 3.89e5T^{2} \)
79 \( 1 - 85.2iT - 4.93e5T^{2} \)
83 \( 1 + 1.25e3iT - 5.71e5T^{2} \)
89 \( 1 - 840. iT - 7.04e5T^{2} \)
97 \( 1 - 770T + 9.12e5T^{2} \)
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

−9.522809680014970238491522997780, −8.880225720182655089451124420749, −8.055399217673525421226197030573, −7.16634931930807539440871260354, −6.12287030607002163339544954083, −5.54863028253765323341382894059, −3.98499029626208439325922812298, −2.84311664757298957907149873368, −2.00787118761106088620349553122, −0.71085950296386130832905792695, 1.48804247753461022025947677208, 2.54405152122883090771710911091, 3.68869052419302128247290635168, 4.66505599035775836346527785678, 5.54973754397508633122840939892, 6.74116513425336993458544233814, 7.69236822617952786521567330738, 8.526837476854205188889422394349, 9.470001207773290080814313171075, 10.13789774556265608653420995093

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