# Properties

 Degree $2$ Conductor $320$ Sign $-0.994 + 0.106i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 19.1·3-s + (55.2 + 8.66i)5-s + 121. i·7-s + 123.·9-s + 298i·11-s − 485.·13-s + (−1.05e3 − 165. i)15-s − 420. i·17-s + 2.57e3i·19-s − 2.32e3i·21-s + 717. i·23-s + (2.97e3 + 956. i)25-s + 2.29e3·27-s − 3.73e3i·29-s + 6.22e3·31-s + ⋯
 L(s)  = 1 − 1.22·3-s + (0.987 + 0.154i)5-s + 0.937i·7-s + 0.506·9-s + 0.742i·11-s − 0.797·13-s + (−1.21 − 0.190i)15-s − 0.353i·17-s + 1.63i·19-s − 1.15i·21-s + 0.282i·23-s + (0.952 + 0.306i)25-s + 0.606·27-s − 0.824i·29-s + 1.16·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $-0.994 + 0.106i$ Motivic weight: $$5$$ Character: $\chi_{320} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :5/2),\ -0.994 + 0.106i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.5366413978$$ $$L(\frac12)$$ $$\approx$$ $$0.5366413978$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (-55.2 - 8.66i)T$$
good3 $$1 + 19.1T + 243T^{2}$$
7 $$1 - 121. iT - 1.68e4T^{2}$$
11 $$1 - 298iT - 1.61e5T^{2}$$
13 $$1 + 485.T + 3.71e5T^{2}$$
17 $$1 + 420. iT - 1.41e6T^{2}$$
19 $$1 - 2.57e3iT - 2.47e6T^{2}$$
23 $$1 - 717. iT - 6.43e6T^{2}$$
29 $$1 + 3.73e3iT - 2.05e7T^{2}$$
31 $$1 - 6.22e3T + 2.86e7T^{2}$$
37 $$1 + 5.72e3T + 6.93e7T^{2}$$
41 $$1 - 1.13e4T + 1.15e8T^{2}$$
43 $$1 + 1.91e4T + 1.47e8T^{2}$$
47 $$1 + 2.28e3iT - 2.29e8T^{2}$$
53 $$1 + 1.72e4T + 4.18e8T^{2}$$
59 $$1 + 1.63e4iT - 7.14e8T^{2}$$
61 $$1 - 4.87e4iT - 8.44e8T^{2}$$
67 $$1 + 8.39e3T + 1.35e9T^{2}$$
71 $$1 + 2.90e4T + 1.80e9T^{2}$$
73 $$1 + 420. iT - 2.07e9T^{2}$$
79 $$1 + 3.11e4T + 3.07e9T^{2}$$
83 $$1 + 1.06e5T + 3.93e9T^{2}$$
89 $$1 + 9.93e4T + 5.58e9T^{2}$$
97 $$1 + 1.80e5iT - 8.58e9T^{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

−11.43549055057487243767374920476, −10.13611864452427091317818852721, −9.839256417750570459537711505991, −8.545736692859866687370256216162, −7.18167564476570847258834593362, −6.10824767473581513102015253841, −5.57298993660533495180672644757, −4.63627020526938312961526069356, −2.70465077125751782100970246009, −1.55158919650606585430643920607, 0.17910271212036626436635804063, 1.16831888350469217130965480641, 2.83314352951830629074152949656, 4.58483398711268216371129150238, 5.31776056834574124469597201183, 6.38061526438215291960934968734, 7.02437210273869480695895171902, 8.504693313299755175461550995563, 9.614231139222544214800991409476, 10.54348732682148822325023206317