# Properties

 Degree $2$ Conductor $25$ Sign $0.447 - 0.894i$ Motivic weight $33$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 1.32e5i·2-s + 1.17e8i·3-s − 9.07e9·4-s − 1.56e13·6-s − 1.34e13i·7-s − 6.49e13i·8-s − 8.25e15·9-s − 7.96e16·11-s − 1.06e18i·12-s + 1.16e18i·13-s + 1.79e18·14-s − 6.93e19·16-s + 3.27e20i·17-s − 1.09e21i·18-s − 2.21e21·19-s + ⋯
 L(s)  = 1 + 1.43i·2-s + 1.57i·3-s − 1.05·4-s − 2.26·6-s − 0.153i·7-s − 0.0815i·8-s − 1.48·9-s − 0.522·11-s − 1.66i·12-s + 0.486i·13-s + 0.219·14-s − 0.939·16-s + 1.63i·17-s − 2.12i·18-s − 1.76·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $0.447 - 0.894i$ Motivic weight: $$33$$ Character: $\chi_{25} (24, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 25,\ (\ :33/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(17)$$ $$\approx$$ $$0.1554568810$$ $$L(\frac12)$$ $$\approx$$ $$0.1554568810$$ $$L(\frac{35}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 - 1.32e5iT - 8.58e9T^{2}$$
3 $$1 - 1.17e8iT - 5.55e15T^{2}$$
7 $$1 + 1.34e13iT - 7.73e27T^{2}$$
11 $$1 + 7.96e16T + 2.32e34T^{2}$$
13 $$1 - 1.16e18iT - 5.75e36T^{2}$$
17 $$1 - 3.27e20iT - 4.02e40T^{2}$$
19 $$1 + 2.21e21T + 1.58e42T^{2}$$
23 $$1 + 3.93e22iT - 8.65e44T^{2}$$
29 $$1 + 1.59e24T + 1.81e48T^{2}$$
31 $$1 - 2.07e24T + 1.64e49T^{2}$$
37 $$1 + 4.76e25iT - 5.63e51T^{2}$$
41 $$1 + 6.39e26T + 1.66e53T^{2}$$
43 $$1 - 2.16e26iT - 8.02e53T^{2}$$
47 $$1 - 7.45e27iT - 1.51e55T^{2}$$
53 $$1 + 1.70e28iT - 7.96e56T^{2}$$
59 $$1 - 5.91e28T + 2.74e58T^{2}$$
61 $$1 - 2.18e29T + 8.23e58T^{2}$$
67 $$1 - 2.21e30iT - 1.82e60T^{2}$$
71 $$1 - 2.87e30T + 1.23e61T^{2}$$
73 $$1 - 3.41e30iT - 3.08e61T^{2}$$
79 $$1 + 5.87e30T + 4.18e62T^{2}$$
83 $$1 - 2.11e31iT - 2.13e63T^{2}$$
89 $$1 - 2.67e32T + 2.13e64T^{2}$$
97 $$1 - 1.07e31iT - 3.65e65T^{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

−12.93630398001934286280995887671, −11.02191227523356351282135475844, −10.23953823405647905559802097126, −8.873192635958507949878270032931, −8.199037631157385659233435605216, −6.62614253619166537709054537862, −5.69068246748444769393577832702, −4.56789966866343438627753729988, −3.92797848348813698030270708406, −2.23514204748033604339471587156, 0.03868770458918721846629333644, 0.69264242345870491518276435380, 1.83697122514945304294950225978, 2.40329039385924943791021453324, 3.47752046297065423527398814332, 5.15442806868231077504372258616, 6.60570279137269026781164154159, 7.61629677366772877574166337947, 8.876747193425340039430806824987, 10.23653302195385441688767680683