# Properties

 Degree $2$ Conductor $8$ Sign $-0.268 + 0.963i$ Motivic weight $17$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (328. − 151. i)2-s + 4.24e3i·3-s + (8.49e4 − 9.98e4i)4-s − 6.63e5i·5-s + (6.45e5 + 1.39e6i)6-s − 1.66e7·7-s + (1.27e7 − 4.57e7i)8-s + 1.11e8·9-s + (−1.00e8 − 2.18e8i)10-s − 1.05e9i·11-s + (4.24e8 + 3.60e8i)12-s + 9.19e7i·13-s + (−5.47e9 + 2.53e9i)14-s + 2.82e9·15-s + (−2.75e9 − 1.69e10i)16-s − 1.98e10·17-s + ⋯
 L(s)  = 1 + (0.907 − 0.419i)2-s + 0.373i·3-s + (0.648 − 0.761i)4-s − 0.760i·5-s + (0.156 + 0.339i)6-s − 1.09·7-s + (0.268 − 0.963i)8-s + 0.860·9-s + (−0.318 − 0.690i)10-s − 1.48i·11-s + (0.284 + 0.242i)12-s + 0.0312i·13-s + (−0.992 + 0.458i)14-s + 0.284·15-s + (−0.160 − 0.987i)16-s − 0.689·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(18-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8$$    =    $$2^{3}$$ Sign: $-0.268 + 0.963i$ Motivic weight: $$17$$ Character: $\chi_{8} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 8,\ (\ :17/2),\ -0.268 + 0.963i)$$

## Particular Values

 $$L(9)$$ $$\approx$$ $$1.64211 - 2.16298i$$ $$L(\frac12)$$ $$\approx$$ $$1.64211 - 2.16298i$$ $$L(\frac{19}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-328. + 151. i)T$$
good3 $$1 - 4.24e3iT - 1.29e8T^{2}$$
5 $$1 + 6.63e5iT - 7.62e11T^{2}$$
7 $$1 + 1.66e7T + 2.32e14T^{2}$$
11 $$1 + 1.05e9iT - 5.05e17T^{2}$$
13 $$1 - 9.19e7iT - 8.65e18T^{2}$$
17 $$1 + 1.98e10T + 8.27e20T^{2}$$
19 $$1 + 8.44e10iT - 5.48e21T^{2}$$
23 $$1 + 2.72e10T + 1.41e23T^{2}$$
29 $$1 - 3.75e12iT - 7.25e24T^{2}$$
31 $$1 - 5.36e12T + 2.25e25T^{2}$$
37 $$1 - 1.92e13iT - 4.56e26T^{2}$$
41 $$1 - 5.42e13T + 2.61e27T^{2}$$
43 $$1 - 3.96e13iT - 5.87e27T^{2}$$
47 $$1 - 4.48e13T + 2.66e28T^{2}$$
53 $$1 - 7.81e14iT - 2.05e29T^{2}$$
59 $$1 + 1.07e15iT - 1.27e30T^{2}$$
61 $$1 + 1.92e15iT - 2.24e30T^{2}$$
67 $$1 + 3.68e15iT - 1.10e31T^{2}$$
71 $$1 + 1.02e16T + 2.96e31T^{2}$$
73 $$1 - 1.25e16T + 4.74e31T^{2}$$
79 $$1 - 3.24e15T + 1.81e32T^{2}$$
83 $$1 - 4.28e15iT - 4.21e32T^{2}$$
89 $$1 - 6.95e16T + 1.37e33T^{2}$$
97 $$1 + 7.74e16T + 5.95e33T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−16.39635780212676505325872844125, −15.62053849916215171617439854204, −13.59473240751302874472901323153, −12.64487619703354681005926113574, −10.85913758134279591519191290380, −9.249176559491100788260481351008, −6.47694610608886868152917513790, −4.72458911276472190758276880665, −3.15373025124775396266202590568, −0.842591543557848077890502433286, 2.33131756261166549602944838397, 4.11754593410667581352617618277, 6.38609495789129591513581979712, 7.37657687486616011084659147523, 10.11127309234066276949016446941, 12.22187575913595672390977010457, 13.26004488889856699467292581841, 14.88134595963994379378501156734, 16.00009749951639998544242524354, 17.76140554465044456079915534332