Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.158 - 0.987i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.9 + 12.3i)2-s + 67.6i·3-s + (207. + 467. i)4-s + 506. i·5-s + (−834. + 1.28e3i)6-s + 300.·7-s + (−1.83e3 + 1.14e4i)8-s + 1.51e4·9-s + (−6.25e3 + 9.61e3i)10-s − 5.52e4i·11-s + (−3.16e4 + 1.40e4i)12-s − 1.60e5i·13-s + (5.69e3 + 3.70e3i)14-s − 3.42e4·15-s + (−1.75e5 + 1.94e5i)16-s + 3.05e5·17-s + ⋯
L(s)  = 1  + (0.838 + 0.545i)2-s + 0.482i·3-s + (0.405 + 0.914i)4-s + 0.362i·5-s + (−0.262 + 0.404i)6-s + 0.0472·7-s + (−0.158 + 0.987i)8-s + 0.767·9-s + (−0.197 + 0.304i)10-s − 1.13i·11-s + (−0.440 + 0.195i)12-s − 1.55i·13-s + (0.0396 + 0.0257i)14-s − 0.174·15-s + (−0.670 + 0.741i)16-s + 0.886·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.158 - 0.987i$
motivic weight  =  \(9\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :9/2),\ 0.158 - 0.987i)$
$L(5)$  $\approx$  $1.75053 + 1.49225i$
$L(\frac12)$  $\approx$  $1.75053 + 1.49225i$
$L(\frac{11}{2})$   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 \( 1 + (-18.9 - 12.3i)T \)
good3 \( 1 - 67.6iT - 1.96e4T^{2} \)
5 \( 1 - 506. iT - 1.95e6T^{2} \)
7 \( 1 - 300.T + 4.03e7T^{2} \)
11 \( 1 + 5.52e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.60e5iT - 1.06e10T^{2} \)
17 \( 1 - 3.05e5T + 1.18e11T^{2} \)
19 \( 1 - 6.34e5iT - 3.22e11T^{2} \)
23 \( 1 + 2.25e6T + 1.80e12T^{2} \)
29 \( 1 + 1.62e6iT - 1.45e13T^{2} \)
31 \( 1 + 2.76e6T + 2.64e13T^{2} \)
37 \( 1 + 6.07e6iT - 1.29e14T^{2} \)
41 \( 1 + 6.28e6T + 3.27e14T^{2} \)
43 \( 1 + 2.76e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.38e7T + 1.11e15T^{2} \)
53 \( 1 - 1.04e8iT - 3.29e15T^{2} \)
59 \( 1 - 3.86e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.10e8iT - 1.16e16T^{2} \)
67 \( 1 + 1.57e8iT - 2.72e16T^{2} \)
71 \( 1 + 1.55e8T + 4.58e16T^{2} \)
73 \( 1 - 2.67e8T + 5.88e16T^{2} \)
79 \( 1 + 2.31e8T + 1.19e17T^{2} \)
83 \( 1 - 5.10e8iT - 1.86e17T^{2} \)
89 \( 1 + 3.91e8T + 3.50e17T^{2} \)
97 \( 1 + 4.08e8T + 7.60e17T^{2} \)
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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

−20.48719607485803655867989622728, −18.40296490361267609579902071647, −16.58946820601144498908836277980, −15.46777968170485745061311987556, −14.10862111965734679987565821519, −12.48127846311781118117282410228, −10.51138259987051860361226554797, −7.899257537411288051599426808184, −5.73111269654623798831699249891, −3.53901561045568655763416667963, 1.70993775660762988295087346178, 4.51827021190430141688253419569, 6.89286082018099205740364667346, 9.782463760608644528696436301100, 11.84367830379921795258069708987, 12.97619325500560247861138662292, 14.46219352638226535918885426283, 16.13065382618078950656735641591, 18.25995941481561914849259291070, 19.56923155250795989963608775582

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