Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.489 - 0.872i$
Motivic weight 13
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (84.9 + 31.3i)2-s + 1.41e3i·3-s + (6.23e3 + 5.31e3i)4-s + 2.38e3i·5-s + (−4.43e4 + 1.20e5i)6-s − 3.17e5·7-s + (3.62e5 + 6.46e5i)8-s − 4.09e5·9-s + (−7.46e4 + 2.02e5i)10-s − 1.40e6i·11-s + (−7.52e6 + 8.82e6i)12-s + 2.51e7i·13-s + (−2.69e7 − 9.93e6i)14-s − 3.37e6·15-s + (1.05e7 + 6.62e7i)16-s + 1.11e8·17-s + ⋯
L(s)  = 1  + (0.938 + 0.345i)2-s + 1.12i·3-s + (0.760 + 0.649i)4-s + 0.0682i·5-s + (−0.387 + 1.05i)6-s − 1.01·7-s + (0.489 + 0.872i)8-s − 0.257·9-s + (−0.0236 + 0.0640i)10-s − 0.239i·11-s + (−0.727 + 0.852i)12-s + 1.44i·13-s + (−0.956 − 0.352i)14-s − 0.0765·15-s + (0.157 + 0.987i)16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(14-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.489 - 0.872i$
motivic weight  =  \(13\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :13/2),\ -0.489 - 0.872i)$
$L(7)$  $\approx$  $1.36405 + 2.32878i$
$L(\frac12)$  $\approx$  $1.36405 + 2.32878i$
$L(\frac{15}{2})$   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 \( 1 + (-84.9 - 31.3i)T \)
good3 \( 1 - 1.41e3iT - 1.59e6T^{2} \)
5 \( 1 - 2.38e3iT - 1.22e9T^{2} \)
7 \( 1 + 3.17e5T + 9.68e10T^{2} \)
11 \( 1 + 1.40e6iT - 3.45e13T^{2} \)
13 \( 1 - 2.51e7iT - 3.02e14T^{2} \)
17 \( 1 - 1.11e8T + 9.90e15T^{2} \)
19 \( 1 + 3.64e8iT - 4.20e16T^{2} \)
23 \( 1 - 7.49e8T + 5.04e17T^{2} \)
29 \( 1 + 2.83e9iT - 1.02e19T^{2} \)
31 \( 1 + 2.93e9T + 2.44e19T^{2} \)
37 \( 1 - 8.46e9iT - 2.43e20T^{2} \)
41 \( 1 - 3.90e10T + 9.25e20T^{2} \)
43 \( 1 - 2.17e10iT - 1.71e21T^{2} \)
47 \( 1 + 1.15e11T + 5.46e21T^{2} \)
53 \( 1 + 1.71e11iT - 2.60e22T^{2} \)
59 \( 1 + 2.45e11iT - 1.04e23T^{2} \)
61 \( 1 + 2.71e11iT - 1.61e23T^{2} \)
67 \( 1 + 6.82e11iT - 5.48e23T^{2} \)
71 \( 1 + 2.90e11T + 1.16e24T^{2} \)
73 \( 1 + 1.48e12T + 1.67e24T^{2} \)
79 \( 1 - 1.58e12T + 4.66e24T^{2} \)
83 \( 1 - 7.79e11iT - 8.87e24T^{2} \)
89 \( 1 + 4.10e12T + 2.19e25T^{2} \)
97 \( 1 - 1.11e13T + 6.73e25T^{2} \)
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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

−19.30689901611971501139712148417, −16.73353960562261827105371308342, −15.95074925626803723076263207401, −14.65430285486327963624262694372, −13.05235596871804076699326235517, −11.21389895642375003055784087042, −9.384146631731733847444242228070, −6.73851022278084696869640654541, −4.75350634760065164536498711230, −3.21749486277340490413237687312, 1.13182754764996151619051846223, 3.15722795232523809558507010116, 5.80726223661423727537792140543, 7.37112245222519883047524485236, 10.26415636584876489398068953832, 12.47239308525618049955048423340, 12.91061189674251290776951684784, 14.61457982478994148036133940801, 16.34137104489828245657127675716, 18.46506258388853030696617146742

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