Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.625 + 0.779i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−749. + 697. i)2-s − 1.61e4·3-s + (7.62e4 − 1.04e6i)4-s + 8.44e6i·5-s + (1.20e7 − 1.12e7i)6-s + 6.17e7i·7-s + (6.72e8 + 8.37e8i)8-s − 3.22e9·9-s + (−5.89e9 − 6.33e9i)10-s − 2.72e10·11-s + (−1.22e9 + 1.68e10i)12-s + 6.05e10i·13-s + (−4.30e10 − 4.62e10i)14-s − 1.36e11i·15-s + (−1.08e12 − 1.59e11i)16-s − 3.07e11·17-s + ⋯
L(s)  = 1  + (−0.732 + 0.680i)2-s − 0.272·3-s + (0.0726 − 0.997i)4-s + 0.865i·5-s + (0.199 − 0.185i)6-s + 0.218i·7-s + (0.625 + 0.779i)8-s − 0.925·9-s + (−0.589 − 0.633i)10-s − 1.05·11-s + (−0.0198 + 0.272i)12-s + 0.439i·13-s + (−0.148 − 0.160i)14-s − 0.236i·15-s + (−0.989 − 0.144i)16-s − 0.152·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.779i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.625 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.625 + 0.779i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ 0.625 + 0.779i)$
$L(\frac{21}{2})$  $\approx$  $0.400597 - 0.192154i$
$L(\frac12)$  $\approx$  $0.400597 - 0.192154i$
$L(11)$   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 \( 1 + (749. - 697. i)T \)
good3 \( 1 + 1.61e4T + 3.48e9T^{2} \)
5 \( 1 - 8.44e6iT - 9.53e13T^{2} \)
7 \( 1 - 6.17e7iT - 7.97e16T^{2} \)
11 \( 1 + 2.72e10T + 6.72e20T^{2} \)
13 \( 1 - 6.05e10iT - 1.90e22T^{2} \)
17 \( 1 + 3.07e11T + 4.06e24T^{2} \)
19 \( 1 - 7.50e12T + 3.75e25T^{2} \)
23 \( 1 + 5.30e13iT - 1.71e27T^{2} \)
29 \( 1 + 5.67e14iT - 1.76e29T^{2} \)
31 \( 1 - 1.32e14iT - 6.71e29T^{2} \)
37 \( 1 - 2.44e14iT - 2.31e31T^{2} \)
41 \( 1 + 1.67e16T + 1.80e32T^{2} \)
43 \( 1 + 8.85e14T + 4.67e32T^{2} \)
47 \( 1 + 7.52e16iT - 2.76e33T^{2} \)
53 \( 1 + 2.98e17iT - 3.05e34T^{2} \)
59 \( 1 + 6.61e16T + 2.61e35T^{2} \)
61 \( 1 + 1.14e17iT - 5.08e35T^{2} \)
67 \( 1 - 1.03e18T + 3.32e36T^{2} \)
71 \( 1 - 4.52e18iT - 1.05e37T^{2} \)
73 \( 1 + 3.78e18T + 1.84e37T^{2} \)
79 \( 1 - 9.29e18iT - 8.96e37T^{2} \)
83 \( 1 - 2.33e19T + 2.40e38T^{2} \)
89 \( 1 + 4.76e19T + 9.72e38T^{2} \)
97 \( 1 + 7.53e19T + 5.43e39T^{2} \)
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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

−16.55882635514953864520783022449, −15.19163202468126302380410157047, −13.94036899814118855818757212150, −11.43329641172551746232700591394, −10.15635743479038945809274261610, −8.371305237770345024341380588679, −6.79870375391410752611156126578, −5.37195219441553213103570208298, −2.53498933115058389980863431139, −0.24415449313720473725729478692, 1.10756194422560305964128884921, 3.05520780793859666105405733149, 5.20285175056676615697437785803, 7.72807329601555137875676077816, 9.099415057950877882376875219501, 10.72105112753345130324376931802, 12.10887165600358609173234255549, 13.44671994362258546452838493593, 15.94729595176152213436961799377, 17.11437168320234740039171522455

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