# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $-0.965 + 0.258i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 16.5i·3-s + 25i·5-s − 55.4·7-s − 31.1·9-s + 153. i·11-s + 57.3i·13-s + 413.·15-s + 1.78e3·17-s − 1.59e3i·19-s + 918. i·21-s − 4.11e3·23-s − 625·25-s − 3.50e3i·27-s − 5.00e3i·29-s − 1.82e3·31-s + ⋯
 L(s)  = 1 − 1.06i·3-s + 0.447i·5-s − 0.427·7-s − 0.128·9-s + 0.382i·11-s + 0.0940i·13-s + 0.475·15-s + 1.49·17-s − 1.01i·19-s + 0.454i·21-s − 1.62·23-s − 0.200·25-s − 0.925i·27-s − 1.10i·29-s − 0.340·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $-0.965 + 0.258i$ motivic weight = $$5$$ character : $\chi_{320} (161, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ -0.965 + 0.258i)$$ $$L(3)$$ $$\approx$$ $$0.9993862735$$ $$L(\frac12)$$ $$\approx$$ $$0.9993862735$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - 25iT$$
good3 $$1 + 16.5iT - 243T^{2}$$
7 $$1 + 55.4T + 1.68e4T^{2}$$
11 $$1 - 153. iT - 1.61e5T^{2}$$
13 $$1 - 57.3iT - 3.71e5T^{2}$$
17 $$1 - 1.78e3T + 1.41e6T^{2}$$
19 $$1 + 1.59e3iT - 2.47e6T^{2}$$
23 $$1 + 4.11e3T + 6.43e6T^{2}$$
29 $$1 + 5.00e3iT - 2.05e7T^{2}$$
31 $$1 + 1.82e3T + 2.86e7T^{2}$$
37 $$1 - 1.80e3iT - 6.93e7T^{2}$$
41 $$1 - 5.45e3T + 1.15e8T^{2}$$
43 $$1 + 8.25e3iT - 1.47e8T^{2}$$
47 $$1 + 8.40e3T + 2.29e8T^{2}$$
53 $$1 + 6.74e3iT - 4.18e8T^{2}$$
59 $$1 + 17.5iT - 7.14e8T^{2}$$
61 $$1 - 1.88e4iT - 8.44e8T^{2}$$
67 $$1 - 5.97e3iT - 1.35e9T^{2}$$
71 $$1 - 1.68e4T + 1.80e9T^{2}$$
73 $$1 + 5.66e4T + 2.07e9T^{2}$$
79 $$1 + 3.72e4T + 3.07e9T^{2}$$
83 $$1 + 5.78e4iT - 3.93e9T^{2}$$
89 $$1 + 9.31e4T + 5.58e9T^{2}$$
97 $$1 + 9.12e4T + 8.58e9T^{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

−10.23930529873486129525501043687, −9.604467405970492152734004193658, −8.156505864127968206193517601330, −7.42842025337638863372704071571, −6.60097563407526832286117773503, −5.69387792845239613347957017798, −4.12184225586284327058626147355, −2.77046799936795646809726219999, −1.62856870754518905643971614165, −0.26395195007646110864636698218, 1.40603603756712659564852137878, 3.26554012045303711294523483816, 4.04592064682862567714265384348, 5.24197053279454247530523240591, 6.07751607676997700517780841570, 7.58572070286525733314143608078, 8.510176055358193198153043337906, 9.686154148378528157012371475686, 10.02196082969068156534199719976, 11.03865797616050680675212079218