↑

Properties

Label 2-2e3-8.3-c18-0-15
Degree $2$
Conductor $8$
Sign $-0.828 + 0.560i$
Analytic cond. $16.4308$
Root an. cond. $4.05350$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (501. − 100. i)2-s − 1.26e4·3-s + (2.41e5 − 1.01e5i)4-s − 2.68e6i·5-s + (−6.35e6 + 1.27e6i)6-s + 4.44e7i·7-s + (1.11e8 − 7.51e7i)8-s − 2.27e8·9-s + (−2.70e8 − 1.34e9i)10-s − 2.94e9·11-s + (−3.06e9 + 1.28e9i)12-s − 2.11e10i·13-s + (4.48e9 + 2.23e10i)14-s + 3.40e10i·15-s + (4.82e10 − 4.89e10i)16-s − 8.60e10·17-s + ⋯
L(s)  = 1  + (0.980 − 0.196i)2-s − 0.643·3-s + (0.922 − 0.385i)4-s − 1.37i·5-s + (−0.630 + 0.126i)6-s + 1.10i·7-s + (0.828 − 0.560i)8-s − 0.585·9-s + (−0.270 − 1.34i)10-s − 1.24·11-s + (−0.593 + 0.248i)12-s − 1.99i·13-s + (0.217 + 1.08i)14-s + 0.884i·15-s + (0.702 − 0.712i)16-s − 0.725·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.828 + 0.560i$
Analytic conductor: \(16.4308\)
Root analytic conductor: \(4.05350\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9),\ -0.828 + 0.560i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.490595 - 1.60185i\)
\(L(\frac12)\) \(\approx\) \(0.490595 - 1.60185i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-501. + 100. i)T \)
good3 \( 1 + 1.26e4T + 3.87e8T^{2} \)
5 \( 1 + 2.68e6iT - 3.81e12T^{2} \)
7 \( 1 - 4.44e7iT - 1.62e15T^{2} \)
11 \( 1 + 2.94e9T + 5.55e18T^{2} \)
13 \( 1 + 2.11e10iT - 1.12e20T^{2} \)
17 \( 1 + 8.60e10T + 1.40e22T^{2} \)
19 \( 1 + 1.59e11T + 1.04e23T^{2} \)
23 \( 1 + 8.84e11iT - 3.24e24T^{2} \)
29 \( 1 - 1.10e13iT - 2.10e26T^{2} \)
31 \( 1 - 9.18e12iT - 6.99e26T^{2} \)
37 \( 1 + 1.80e13iT - 1.68e28T^{2} \)
41 \( 1 - 4.66e14T + 1.07e29T^{2} \)
43 \( 1 - 1.02e14T + 2.52e29T^{2} \)
47 \( 1 + 1.49e15iT - 1.25e30T^{2} \)
53 \( 1 + 4.93e14iT - 1.08e31T^{2} \)
59 \( 1 - 1.08e16T + 7.50e31T^{2} \)
61 \( 1 + 7.51e15iT - 1.36e32T^{2} \)
67 \( 1 + 1.79e16T + 7.40e32T^{2} \)
71 \( 1 + 4.48e16iT - 2.10e33T^{2} \)
73 \( 1 + 5.61e16T + 3.46e33T^{2} \)
79 \( 1 - 2.04e17iT - 1.43e34T^{2} \)
83 \( 1 - 2.79e17T + 3.49e34T^{2} \)
89 \( 1 + 3.39e17T + 1.22e35T^{2} \)
97 \( 1 + 3.43e17T + 5.77e35T^{2} \)
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

−16.25113358704537358187652629612, −15.22110690565245785731165676370, −12.99493042475851067920758597988, −12.35883754578678141900181688443, −10.73069673025293773200126515509, −8.393088108631860802567089651289, −5.68097062719934162485249392923, −5.07291012687500470123035991281, −2.61448918840245710253573785350, −0.47466323897001560925312014765, 2.50163307692574958742396674879, 4.27274283273300805177121481161, 6.22954493633937034330687611788, 7.31495403375013780180478292554, 10.67994683465450187848322411452, 11.45271983155175317562980920386, 13.53633526571090358870240811926, 14.54532298773573392071694134005, 16.17096873900238394065976689039, 17.49026965387126015139706501781

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