Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.372 - 0.928i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.86 − 22.4i)2-s + 247. i·3-s + (−495. − 128. i)4-s + 1.41e3i·5-s + (5.55e3 + 709. i)6-s + 5.08e3·7-s + (−4.31e3 + 1.07e4i)8-s − 4.15e4·9-s + (3.18e4 + 4.06e3i)10-s − 1.48e4i·11-s + (3.18e4 − 1.22e5i)12-s + 6.40e4i·13-s + (1.45e4 − 1.14e5i)14-s − 3.50e5·15-s + (2.28e5 + 1.27e5i)16-s + 2.51e5·17-s + ⋯
L(s)  = 1  + (0.126 − 0.991i)2-s + 1.76i·3-s + (−0.967 − 0.251i)4-s + 1.01i·5-s + (1.74 + 0.223i)6-s + 0.800·7-s + (−0.372 + 0.928i)8-s − 2.10·9-s + (1.00 + 0.128i)10-s − 0.305i·11-s + (0.443 − 1.70i)12-s + 0.622i·13-s + (0.101 − 0.794i)14-s − 1.78·15-s + (0.873 + 0.486i)16-s + 0.729·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.372 - 0.928i$
motivic weight  =  \(9\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :9/2),\ 0.372 - 0.928i)$
$L(5)$  $\approx$  $1.14076 + 0.771520i$
$L(\frac12)$  $\approx$  $1.14076 + 0.771520i$
$L(\frac{11}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-2.86 + 22.4i)T \)
good3 \( 1 - 247. iT - 1.96e4T^{2} \)
5 \( 1 - 1.41e3iT - 1.95e6T^{2} \)
7 \( 1 - 5.08e3T + 4.03e7T^{2} \)
11 \( 1 + 1.48e4iT - 2.35e9T^{2} \)
13 \( 1 - 6.40e4iT - 1.06e10T^{2} \)
17 \( 1 - 2.51e5T + 1.18e11T^{2} \)
19 \( 1 + 5.11e5iT - 3.22e11T^{2} \)
23 \( 1 - 1.96e6T + 1.80e12T^{2} \)
29 \( 1 - 2.16e6iT - 1.45e13T^{2} \)
31 \( 1 + 3.03e6T + 2.64e13T^{2} \)
37 \( 1 - 8.74e6iT - 1.29e14T^{2} \)
41 \( 1 + 1.48e7T + 3.27e14T^{2} \)
43 \( 1 - 1.53e5iT - 5.02e14T^{2} \)
47 \( 1 - 5.17e7T + 1.11e15T^{2} \)
53 \( 1 - 4.26e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.15e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.95e8iT - 1.16e16T^{2} \)
67 \( 1 + 1.69e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.23e8T + 4.58e16T^{2} \)
73 \( 1 + 4.10e8T + 5.88e16T^{2} \)
79 \( 1 - 2.51e7T + 1.19e17T^{2} \)
83 \( 1 + 4.73e8iT - 1.86e17T^{2} \)
89 \( 1 - 8.45e8T + 3.50e17T^{2} \)
97 \( 1 + 7.97e8T + 7.60e17T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−20.40902333459441252962047641186, −18.75742403191153948371137546485, −17.07247912318096439514229388370, −15.10227852350045617971139098218, −14.19248293819079597878766064704, −11.34888803696935809559397074830, −10.57606590784860215865387190350, −9.066119657328896677898637071310, −4.91200434140504340858827066423, −3.19765859938353322157558525430, 1.02553271768728285703819471725, 5.47507876430754981413015897677, 7.42601265088147744082964880155, 8.539255450859165808688795095002, 12.26000126045001234446110840509, 13.20262344988885223296115608556, 14.62244718858995228697031701352, 16.81593882016075729106360703200, 17.78133996042136960815267228662, 18.96007466625238273116851262794

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