Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.303 - 0.952i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (829. − 600. i)2-s − 1.03e5·3-s + (3.28e5 − 9.95e5i)4-s − 1.04e7i·5-s + (−8.56e7 + 6.19e7i)6-s − 3.68e8i·7-s + (−3.25e8 − 1.02e9i)8-s + 7.17e9·9-s + (−6.25e9 − 8.65e9i)10-s − 3.35e10·11-s + (−3.38e10 + 1.02e11i)12-s + 1.95e11i·13-s + (−2.20e11 − 3.05e11i)14-s + 1.07e12i·15-s + (−8.84e11 − 6.53e11i)16-s − 1.59e11·17-s + ⋯
L(s)  = 1  + (0.810 − 0.586i)2-s − 1.74·3-s + (0.312 − 0.949i)4-s − 1.06i·5-s + (−1.41 + 1.02i)6-s − 1.30i·7-s + (−0.303 − 0.952i)8-s + 2.05·9-s + (−0.625 − 0.865i)10-s − 1.29·11-s + (−0.547 + 1.66i)12-s + 1.41i·13-s + (−0.763 − 1.05i)14-s + 1.86i·15-s + (−0.804 − 0.594i)16-s − 0.0793·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.303 - 0.952i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ -0.303 - 0.952i)$
$L(\frac{21}{2})$  $\approx$  $0.375747 + 0.513863i$
$L(\frac12)$  $\approx$  $0.375747 + 0.513863i$
$L(11)$   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 \( 1 + (-829. + 600. i)T \)
good3 \( 1 + 1.03e5T + 3.48e9T^{2} \)
5 \( 1 + 1.04e7iT - 9.53e13T^{2} \)
7 \( 1 + 3.68e8iT - 7.97e16T^{2} \)
11 \( 1 + 3.35e10T + 6.72e20T^{2} \)
13 \( 1 - 1.95e11iT - 1.90e22T^{2} \)
17 \( 1 + 1.59e11T + 4.06e24T^{2} \)
19 \( 1 - 4.90e12T + 3.75e25T^{2} \)
23 \( 1 + 3.43e13iT - 1.71e27T^{2} \)
29 \( 1 - 1.07e14iT - 1.76e29T^{2} \)
31 \( 1 - 3.09e13iT - 6.71e29T^{2} \)
37 \( 1 - 4.26e15iT - 2.31e31T^{2} \)
41 \( 1 + 3.78e15T + 1.80e32T^{2} \)
43 \( 1 + 2.30e15T + 4.67e32T^{2} \)
47 \( 1 + 3.45e16iT - 2.76e33T^{2} \)
53 \( 1 + 3.51e16iT - 3.05e34T^{2} \)
59 \( 1 - 2.54e17T + 2.61e35T^{2} \)
61 \( 1 + 1.01e18iT - 5.08e35T^{2} \)
67 \( 1 + 2.98e18T + 3.32e36T^{2} \)
71 \( 1 - 8.39e17iT - 1.05e37T^{2} \)
73 \( 1 - 3.32e18T + 1.84e37T^{2} \)
79 \( 1 - 1.22e19iT - 8.96e37T^{2} \)
83 \( 1 + 2.23e19T + 2.40e38T^{2} \)
89 \( 1 + 3.15e19T + 9.72e38T^{2} \)
97 \( 1 - 8.67e19T + 5.43e39T^{2} \)
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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

−16.07318822653444704669849585554, −13.56835195533398141604762256804, −12.45465371835856762145322279727, −11.23390556781474339036887008664, −10.09727228898410589694095300859, −6.83256547070420524536214982124, −5.19330523887985485081936118348, −4.37106213955288779161412043497, −1.29269493636689429663224909885, −0.23457097092372126837501991018, 2.83691087912140014563343018124, 5.27828643041458613971628151490, 5.90599650144326345097148413655, 7.48363393251154628432637411800, 10.60028628254069796093857395206, 11.79082622729443682302049678971, 12.96850959520335880033608965608, 15.21246183422829996194672315468, 15.93807465536935287920595131771, 17.74790295503015967241173140633

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