Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.634 - 0.772i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (979. − 297. i)2-s − 1.18e4·3-s + (8.72e5 − 5.82e5i)4-s + 4.86e6i·5-s + (−1.16e7 + 3.51e6i)6-s + 3.96e8i·7-s + (6.81e8 − 8.29e8i)8-s − 3.34e9·9-s + (1.44e9 + 4.76e9i)10-s + 2.78e10·11-s + (−1.03e10 + 6.89e9i)12-s + 1.92e11i·13-s + (1.17e11 + 3.88e11i)14-s − 5.75e10i·15-s + (4.21e11 − 1.01e12i)16-s + 1.00e12·17-s + ⋯
L(s)  = 1  + (0.956 − 0.290i)2-s − 0.200·3-s + (0.831 − 0.555i)4-s + 0.497i·5-s + (−0.191 + 0.0581i)6-s + 1.40i·7-s + (0.634 − 0.772i)8-s − 0.959·9-s + (0.144 + 0.476i)10-s + 1.07·11-s + (−0.166 + 0.111i)12-s + 1.39i·13-s + (0.407 + 1.34i)14-s − 0.0997i·15-s + (0.383 − 0.923i)16-s + 0.499·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.634 - 0.772i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ 0.634 - 0.772i)$
$L(\frac{21}{2})$  $\approx$  $2.78843 + 1.31819i$
$L(\frac12)$  $\approx$  $2.78843 + 1.31819i$
$L(11)$   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 \( 1 + (-979. + 297. i)T \)
good3 \( 1 + 1.18e4T + 3.48e9T^{2} \)
5 \( 1 - 4.86e6iT - 9.53e13T^{2} \)
7 \( 1 - 3.96e8iT - 7.97e16T^{2} \)
11 \( 1 - 2.78e10T + 6.72e20T^{2} \)
13 \( 1 - 1.92e11iT - 1.90e22T^{2} \)
17 \( 1 - 1.00e12T + 4.06e24T^{2} \)
19 \( 1 - 9.04e11T + 3.75e25T^{2} \)
23 \( 1 - 5.92e13iT - 1.71e27T^{2} \)
29 \( 1 - 4.75e14iT - 1.76e29T^{2} \)
31 \( 1 + 5.26e14iT - 6.71e29T^{2} \)
37 \( 1 + 5.41e15iT - 2.31e31T^{2} \)
41 \( 1 + 1.00e16T + 1.80e32T^{2} \)
43 \( 1 + 2.73e16T + 4.67e32T^{2} \)
47 \( 1 + 5.52e16iT - 2.76e33T^{2} \)
53 \( 1 + 1.63e16iT - 3.05e34T^{2} \)
59 \( 1 + 5.08e17T + 2.61e35T^{2} \)
61 \( 1 + 2.76e17iT - 5.08e35T^{2} \)
67 \( 1 + 7.08e17T + 3.32e36T^{2} \)
71 \( 1 - 2.20e18iT - 1.05e37T^{2} \)
73 \( 1 - 8.31e18T + 1.84e37T^{2} \)
79 \( 1 - 2.46e18iT - 8.96e37T^{2} \)
83 \( 1 - 1.22e19T + 2.40e38T^{2} \)
89 \( 1 - 3.83e19T + 9.72e38T^{2} \)
97 \( 1 - 2.89e19T + 5.43e39T^{2} \)
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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

−16.62608923201308037563752911960, −15.00854669594668361912215139150, −14.08336174007711120196400078851, −12.07559206130902104382870851866, −11.36426020528402585362038878200, −9.188482972571520476361677229307, −6.62242240954905742710093701195, −5.37442657679465697098544763608, −3.36292322096872594602036836423, −1.86696626341114144736833152318, 0.865791374022349318508394261388, 3.28307718132646410500967680402, 4.80414787452927385511861216711, 6.43822753256554151238768600859, 8.117923937290106872628002052347, 10.62906001618591190095173030897, 12.15350407812336930329841709422, 13.58288048444773420561778921863, 14.73975204487199664759642106522, 16.61366807737597527685511934660

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