Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.573 - 0.819i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−21.4 + 7.11i)2-s − 100. i·3-s + (410. − 305. i)4-s + 2.58e3i·5-s + (715. + 2.15e3i)6-s + 6.96e3·7-s + (−6.64e3 + 9.49e3i)8-s + 9.58e3·9-s + (−1.83e4 − 5.54e4i)10-s + 2.50e4i·11-s + (−3.07e4 − 4.12e4i)12-s + 2.97e4i·13-s + (−1.49e5 + 4.96e4i)14-s + 2.59e5·15-s + (7.50e4 − 2.51e5i)16-s − 1.38e5·17-s + ⋯
L(s)  = 1  + (−0.949 + 0.314i)2-s − 0.716i·3-s + (0.802 − 0.597i)4-s + 1.84i·5-s + (0.225 + 0.679i)6-s + 1.09·7-s + (−0.573 + 0.819i)8-s + 0.487·9-s + (−0.581 − 1.75i)10-s + 0.515i·11-s + (−0.427 − 0.574i)12-s + 0.288i·13-s + (−1.04 + 0.345i)14-s + 1.32·15-s + (0.286 − 0.958i)16-s − 0.402·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.573 - 0.819i$
motivic weight  =  \(9\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :9/2),\ 0.573 - 0.819i)$
$L(5)$  $\approx$  $0.945165 + 0.492205i$
$L(\frac12)$  $\approx$  $0.945165 + 0.492205i$
$L(\frac{11}{2})$   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 \( 1 + (21.4 - 7.11i)T \)
good3 \( 1 + 100. iT - 1.96e4T^{2} \)
5 \( 1 - 2.58e3iT - 1.95e6T^{2} \)
7 \( 1 - 6.96e3T + 4.03e7T^{2} \)
11 \( 1 - 2.50e4iT - 2.35e9T^{2} \)
13 \( 1 - 2.97e4iT - 1.06e10T^{2} \)
17 \( 1 + 1.38e5T + 1.18e11T^{2} \)
19 \( 1 - 4.89e5iT - 3.22e11T^{2} \)
23 \( 1 - 8.47e5T + 1.80e12T^{2} \)
29 \( 1 + 1.13e6iT - 1.45e13T^{2} \)
31 \( 1 - 4.35e6T + 2.64e13T^{2} \)
37 \( 1 - 5.35e5iT - 1.29e14T^{2} \)
41 \( 1 - 1.45e7T + 3.27e14T^{2} \)
43 \( 1 + 3.96e7iT - 5.02e14T^{2} \)
47 \( 1 + 4.48e7T + 1.11e15T^{2} \)
53 \( 1 + 4.85e7iT - 3.29e15T^{2} \)
59 \( 1 + 4.19e6iT - 8.66e15T^{2} \)
61 \( 1 - 6.38e7iT - 1.16e16T^{2} \)
67 \( 1 + 5.79e7iT - 2.72e16T^{2} \)
71 \( 1 - 2.74e8T + 4.58e16T^{2} \)
73 \( 1 + 9.16e7T + 5.88e16T^{2} \)
79 \( 1 - 2.02e8T + 1.19e17T^{2} \)
83 \( 1 + 6.11e8iT - 1.86e17T^{2} \)
89 \( 1 + 7.71e8T + 3.50e17T^{2} \)
97 \( 1 - 1.08e9T + 7.60e17T^{2} \)
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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

−19.22784142195349981092982121380, −18.36835656319087843512172151796, −17.59265689146551637553084060215, −15.30567701247880515374515543179, −14.25351163823008584847309310824, −11.54907983190370644338294532861, −10.25473457542865726636406135738, −7.73811334198341082897571137608, −6.65862218504348266853261555250, −1.99593194959048974090888365981, 1.12264873584756363001482525246, 4.70123618930792420651478544594, 8.216668033323703049741391489157, 9.382440236462426885343215626529, 11.23106367350128430935763339018, 12.87492135249363602440412513661, 15.57179522050983081870439447032, 16.63272912679831855141749920245, 17.76940791903673486829667626244, 19.73453957329879513740345450452

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