Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.745 - 0.666i$
Motivic weight 13
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−25.0 − 86.9i)2-s − 1.74e3i·3-s + (−6.93e3 + 4.35e3i)4-s − 6.49e4i·5-s + (−1.51e5 + 4.37e4i)6-s + 2.01e5·7-s + (5.52e5 + 4.94e5i)8-s − 1.45e6·9-s + (−5.64e6 + 1.62e6i)10-s + 1.75e6i·11-s + (7.61e6 + 1.21e7i)12-s + 1.36e7i·13-s + (−5.04e6 − 1.75e7i)14-s − 1.13e8·15-s + (2.91e7 − 6.04e7i)16-s + 1.14e8·17-s + ⋯
L(s)  = 1  + (−0.276 − 0.960i)2-s − 1.38i·3-s + (−0.846 + 0.532i)4-s − 1.85i·5-s + (−1.32 + 0.382i)6-s + 0.646·7-s + (0.745 + 0.666i)8-s − 0.912·9-s + (−1.78 + 0.514i)10-s + 0.299i·11-s + (0.735 + 1.17i)12-s + 0.781i·13-s + (−0.178 − 0.621i)14-s − 2.56·15-s + (0.433 − 0.901i)16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(14-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.745 - 0.666i$
motivic weight  =  \(13\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :13/2),\ -0.745 - 0.666i)$
$L(7)$  $\approx$  $0.461516 + 1.20922i$
$L(\frac12)$  $\approx$  $0.461516 + 1.20922i$
$L(\frac{15}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (25.0 + 86.9i)T \)
good3 \( 1 + 1.74e3iT - 1.59e6T^{2} \)
5 \( 1 + 6.49e4iT - 1.22e9T^{2} \)
7 \( 1 - 2.01e5T + 9.68e10T^{2} \)
11 \( 1 - 1.75e6iT - 3.45e13T^{2} \)
13 \( 1 - 1.36e7iT - 3.02e14T^{2} \)
17 \( 1 - 1.14e8T + 9.90e15T^{2} \)
19 \( 1 + 2.75e8iT - 4.20e16T^{2} \)
23 \( 1 - 1.08e8T + 5.04e17T^{2} \)
29 \( 1 - 1.20e9iT - 1.02e19T^{2} \)
31 \( 1 + 1.80e9T + 2.44e19T^{2} \)
37 \( 1 - 1.90e10iT - 2.43e20T^{2} \)
41 \( 1 - 2.47e10T + 9.25e20T^{2} \)
43 \( 1 + 1.71e10iT - 1.71e21T^{2} \)
47 \( 1 - 1.08e10T + 5.46e21T^{2} \)
53 \( 1 - 8.06e10iT - 2.60e22T^{2} \)
59 \( 1 - 4.80e10iT - 1.04e23T^{2} \)
61 \( 1 + 5.26e10iT - 1.61e23T^{2} \)
67 \( 1 + 1.29e12iT - 5.48e23T^{2} \)
71 \( 1 + 1.26e12T + 1.16e24T^{2} \)
73 \( 1 - 6.89e11T + 1.67e24T^{2} \)
79 \( 1 - 3.83e12T + 4.66e24T^{2} \)
83 \( 1 + 9.29e11iT - 8.87e24T^{2} \)
89 \( 1 - 6.60e12T + 2.19e25T^{2} \)
97 \( 1 - 8.12e11T + 6.73e25T^{2} \)
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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

−17.82121379116344986896430686148, −16.78198085782100675187110491447, −13.70115765583539007798010501375, −12.64853015411180759604420201743, −11.77096500495012810237503163847, −9.121740731402728551771484134080, −7.84648672680549767942940874540, −4.82721055450893994686983548483, −1.74098504658635278114485834352, −0.78765903334435241606060204547, 3.63292485695676890412364927162, 5.72427749505962053177887307845, 7.73480261796087694801682201245, 9.928394416061576190790644866503, 10.81201254050644151406573856045, 14.39583429591533696613293166773, 14.87776562277828463120617636441, 16.19536151295133724952266402951, 17.79905639530633217084333151559, 18.97395826730268465260281312263

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