Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.944 + 0.328i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (607. + 824. i)2-s + 2.43e4·3-s + (−3.10e5 + 1.00e6i)4-s − 1.39e7i·5-s + (1.47e7 + 2.00e7i)6-s + 2.84e8i·7-s + (−1.01e9 + 3.52e8i)8-s − 2.89e9·9-s + (1.15e10 − 8.48e9i)10-s − 4.00e10·11-s + (−7.54e9 + 2.43e10i)12-s + 2.05e11i·13-s + (−2.34e11 + 1.72e11i)14-s − 3.39e11i·15-s + (−9.06e11 − 6.21e11i)16-s + 2.13e12·17-s + ⋯
L(s)  = 1  + (0.593 + 0.804i)2-s + 0.411·3-s + (−0.295 + 0.955i)4-s − 1.43i·5-s + (0.244 + 0.331i)6-s + 1.00i·7-s + (−0.944 + 0.328i)8-s − 0.830·9-s + (1.15 − 0.848i)10-s − 1.54·11-s + (−0.121 + 0.393i)12-s + 1.49i·13-s + (−0.810 + 0.597i)14-s − 0.588i·15-s + (−0.824 − 0.565i)16-s + 1.05·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.944 + 0.328i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ -0.944 + 0.328i)$
$L(\frac{21}{2})$  $\approx$  $0.140332 - 0.830521i$
$L(\frac12)$  $\approx$  $0.140332 - 0.830521i$
$L(11)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-607. - 824. i)T \)
good3 \( 1 - 2.43e4T + 3.48e9T^{2} \)
5 \( 1 + 1.39e7iT - 9.53e13T^{2} \)
7 \( 1 - 2.84e8iT - 7.97e16T^{2} \)
11 \( 1 + 4.00e10T + 6.72e20T^{2} \)
13 \( 1 - 2.05e11iT - 1.90e22T^{2} \)
17 \( 1 - 2.13e12T + 4.06e24T^{2} \)
19 \( 1 + 8.51e12T + 3.75e25T^{2} \)
23 \( 1 + 3.75e13iT - 1.71e27T^{2} \)
29 \( 1 + 8.26e13iT - 1.76e29T^{2} \)
31 \( 1 - 1.32e14iT - 6.71e29T^{2} \)
37 \( 1 - 5.16e15iT - 2.31e31T^{2} \)
41 \( 1 - 1.72e16T + 1.80e32T^{2} \)
43 \( 1 + 7.35e15T + 4.67e32T^{2} \)
47 \( 1 - 2.83e16iT - 2.76e33T^{2} \)
53 \( 1 + 2.16e16iT - 3.05e34T^{2} \)
59 \( 1 + 2.04e17T + 2.61e35T^{2} \)
61 \( 1 - 6.34e17iT - 5.08e35T^{2} \)
67 \( 1 - 5.81e17T + 3.32e36T^{2} \)
71 \( 1 - 1.90e18iT - 1.05e37T^{2} \)
73 \( 1 + 6.95e18T + 1.84e37T^{2} \)
79 \( 1 + 3.49e18iT - 8.96e37T^{2} \)
83 \( 1 + 1.42e19T + 2.40e38T^{2} \)
89 \( 1 - 2.96e19T + 9.72e38T^{2} \)
97 \( 1 - 6.85e19T + 5.43e39T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.98115503234448695836919960417, −16.02009992750099829690195500392, −14.59860506998235114900428207529, −13.10356533002974015753386072977, −12.04246905213599252987661784020, −8.958844970038191999655688903001, −8.184731120521377928240790990395, −5.81799175888152229784067930141, −4.62568370135661421732631697048, −2.49784556041202800474571492228, 0.21647337150744613969396672561, 2.56326813991890541643262161232, 3.44942643236656469746367117637, 5.68511009250507720615061291409, 7.71059643519921381080620394966, 10.29353060600850119508080676743, 10.89109182218321975574792115789, 13.02666228950224767894092739929, 14.24565105341871763713500926317, 15.22596183288137929706195562401

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