# Properties

 Degree 2 Conductor $2^{3}$ Sign $0.114 + 0.993i$ Motivic weight 20 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−39.1 − 1.02e3i)2-s − 6.15e4·3-s + (−1.04e6 + 8.00e4i)4-s + 2.85e6i·5-s + (2.40e6 + 6.29e7i)6-s + 1.54e8i·7-s + (1.22e8 + 1.06e9i)8-s + 3.02e8·9-s + (2.92e9 − 1.11e8i)10-s + 1.20e10·11-s + (6.43e10 − 4.92e9i)12-s − 1.32e10i·13-s + (1.58e11 − 6.05e9i)14-s − 1.75e11i·15-s + (1.08e12 − 1.67e11i)16-s − 1.51e12·17-s + ⋯
 L(s)  = 1 + (−0.0382 − 0.999i)2-s − 1.04·3-s + (−0.997 + 0.0763i)4-s + 0.292i·5-s + (0.0398 + 1.04i)6-s + 0.547i·7-s + (0.114 + 0.993i)8-s + 0.0868·9-s + (0.292 − 0.0111i)10-s + 0.465·11-s + (1.03 − 0.0796i)12-s − 0.0961i·13-s + (0.547 − 0.0209i)14-s − 0.305i·15-s + (0.988 − 0.152i)16-s − 0.752·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(21-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $0.114 + 0.993i$ motivic weight = $$20$$ character : $\chi_{8} (3, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 8,\ (\ :10),\ 0.114 + 0.993i)$ $L(\frac{21}{2})$ $\approx$ $0.638504 - 0.569188i$ $L(\frac12)$ $\approx$ $0.638504 - 0.569188i$ $L(11)$ 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 + (39.1 + 1.02e3i)T$$
good3 $$1 + 6.15e4T + 3.48e9T^{2}$$
5 $$1 - 2.85e6iT - 9.53e13T^{2}$$
7 $$1 - 1.54e8iT - 7.97e16T^{2}$$
11 $$1 - 1.20e10T + 6.72e20T^{2}$$
13 $$1 + 1.32e10iT - 1.90e22T^{2}$$
17 $$1 + 1.51e12T + 4.06e24T^{2}$$
19 $$1 + 5.95e12T + 3.75e25T^{2}$$
23 $$1 + 5.68e13iT - 1.71e27T^{2}$$
29 $$1 - 9.55e13iT - 1.76e29T^{2}$$
31 $$1 + 1.17e15iT - 6.71e29T^{2}$$
37 $$1 - 8.09e15iT - 2.31e31T^{2}$$
41 $$1 - 1.78e16T + 1.80e32T^{2}$$
43 $$1 - 2.76e16T + 4.67e32T^{2}$$
47 $$1 - 1.55e16iT - 2.76e33T^{2}$$
53 $$1 + 2.45e17iT - 3.05e34T^{2}$$
59 $$1 - 4.21e17T + 2.61e35T^{2}$$
61 $$1 - 5.24e17iT - 5.08e35T^{2}$$
67 $$1 + 2.46e17T + 3.32e36T^{2}$$
71 $$1 + 3.30e18iT - 1.05e37T^{2}$$
73 $$1 + 3.00e18T + 1.84e37T^{2}$$
79 $$1 + 1.13e19iT - 8.96e37T^{2}$$
83 $$1 + 4.84e18T + 2.40e38T^{2}$$
89 $$1 - 1.43e19T + 9.72e38T^{2}$$
97 $$1 - 2.08e18T + 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

−16.80039217786022184738245743026, −14.67587090340326542426544558283, −12.80059788991009010129645009826, −11.59658290205360089368221557948, −10.56798803784212168739874858456, −8.771524835211925689591798996619, −6.18136812910242702705142384780, −4.53584990304748603445082281753, −2.48616065117148326012596020780, −0.56480692835002547254947116101, 0.803367619662063441917762232673, 4.28802188612065284077129417342, 5.72490247754426836590423496212, 7.04900349219617389750887897285, 8.943805932464655852674813241842, 10.83286399104344739309753096758, 12.63668830702007075911798144482, 14.21416932086920539594218445264, 15.90975038975236702776460227599, 17.04364269400868273423305556579