# Properties

 Degree $2$ Conductor $128$ Sign $-0.258 + 0.966i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.54 + 5.54i)3-s + (−21.7 + 21.7i)5-s − 6.62·7-s + 19.6i·9-s + (90.9 + 90.9i)11-s + (−221. − 221. i)13-s − 240. i·15-s − 132.·17-s + (402. − 402. i)19-s + (36.6 − 36.6i)21-s + 27.5·23-s − 320. i·25-s + (−557. − 557. i)27-s + (−174. − 174. i)29-s − 1.08e3i·31-s + ⋯
 L(s)  = 1 + (−0.615 + 0.615i)3-s + (−0.869 + 0.869i)5-s − 0.135·7-s + 0.242i·9-s + (0.752 + 0.752i)11-s + (−1.31 − 1.31i)13-s − 1.07i·15-s − 0.458·17-s + (1.11 − 1.11i)19-s + (0.0831 − 0.0831i)21-s + 0.0519·23-s − 0.512i·25-s + (−0.764 − 0.764i)27-s + (−0.207 − 0.207i)29-s − 1.12i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $-0.258 + 0.966i$ Motivic weight: $$4$$ Character: $\chi_{128} (95, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :2),\ -0.258 + 0.966i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.0679138 - 0.0884764i$$ $$L(\frac12)$$ $$\approx$$ $$0.0679138 - 0.0884764i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + (5.54 - 5.54i)T - 81iT^{2}$$
5 $$1 + (21.7 - 21.7i)T - 625iT^{2}$$
7 $$1 + 6.62T + 2.40e3T^{2}$$
11 $$1 + (-90.9 - 90.9i)T + 1.46e4iT^{2}$$
13 $$1 + (221. + 221. i)T + 2.85e4iT^{2}$$
17 $$1 + 132.T + 8.35e4T^{2}$$
19 $$1 + (-402. + 402. i)T - 1.30e5iT^{2}$$
23 $$1 - 27.5T + 2.79e5T^{2}$$
29 $$1 + (174. + 174. i)T + 7.07e5iT^{2}$$
31 $$1 + 1.08e3iT - 9.23e5T^{2}$$
37 $$1 + (553. - 553. i)T - 1.87e6iT^{2}$$
41 $$1 + 1.80e3iT - 2.82e6T^{2}$$
43 $$1 + (17.8 + 17.8i)T + 3.41e6iT^{2}$$
47 $$1 - 2.26e3iT - 4.87e6T^{2}$$
53 $$1 + (-822. + 822. i)T - 7.89e6iT^{2}$$
59 $$1 + (-972. - 972. i)T + 1.21e7iT^{2}$$
61 $$1 + (-2.05e3 - 2.05e3i)T + 1.38e7iT^{2}$$
67 $$1 + (4.61e3 - 4.61e3i)T - 2.01e7iT^{2}$$
71 $$1 + 3.10e3T + 2.54e7T^{2}$$
73 $$1 - 723. iT - 2.83e7T^{2}$$
79 $$1 + 3.41e3iT - 3.89e7T^{2}$$
83 $$1 + (-161. + 161. i)T - 4.74e7iT^{2}$$
89 $$1 + 1.46e3iT - 6.27e7T^{2}$$
97 $$1 + 8.26e3T + 8.85e7T^{2}$$
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$$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

−12.00819432058152465268662513442, −11.31437657706937049536312153643, −10.35965120533682101056278081851, −9.492762317498940406094133116151, −7.73115604712568516298142759366, −7.00258007191260442316354479038, −5.38240600311613595704135025001, −4.26690853838354810058213686692, −2.78274516930953647343859368252, −0.05395311701507841518837322622, 1.34469767202634270581938669457, 3.71243605080742447824123777291, 5.02426396578805259490191492777, 6.43630804248601222233711421257, 7.40617274414412821509835547581, 8.691656338329548305127196243927, 9.638066046967371948692635970059, 11.38299415777482512385902019723, 12.00031448857989498240111486328, 12.50796311137591251369280296135