# Properties

 Degree $2$ Conductor $5$ Sign $0.992 + 0.124i$ Motivic weight $32$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−8.66e4 − 8.66e4i)2-s + (3.32e7 − 3.32e7i)3-s + 1.07e10i·4-s + (1.18e11 + 9.57e10i)5-s − 5.76e12·6-s + (−3.06e13 − 3.06e13i)7-s + (5.57e14 − 5.57e14i)8-s − 3.55e14i·9-s + (−2.00e15 − 1.85e16i)10-s − 5.53e16·11-s + (3.56e17 + 3.56e17i)12-s + (1.61e17 − 1.61e17i)13-s + 5.31e18i·14-s + (7.12e18 − 7.66e17i)15-s − 5.06e19·16-s + (−2.56e19 − 2.56e19i)17-s + ⋯
 L(s)  = 1 + (−1.32 − 1.32i)2-s + (0.771 − 0.771i)3-s + 2.49i·4-s + (0.778 + 0.627i)5-s − 2.04·6-s + (−0.922 − 0.922i)7-s + (1.98 − 1.98i)8-s − 0.191i·9-s + (−0.200 − 1.85i)10-s − 1.20·11-s + (1.92 + 1.92i)12-s + (0.243 − 0.243i)13-s + 2.43i·14-s + (1.08 − 0.116i)15-s − 2.74·16-s + (−0.527 − 0.527i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(33-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5$$ Sign: $0.992 + 0.124i$ Motivic weight: $$32$$ Character: $\chi_{5} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5,\ (\ :16),\ 0.992 + 0.124i)$$

## Particular Values

 $$L(\frac{33}{2})$$ $$\approx$$ $$0.7719159706$$ $$L(\frac12)$$ $$\approx$$ $$0.7719159706$$ $$L(17)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (-1.18e11 - 9.57e10i)T$$
good2 $$1 + (8.66e4 + 8.66e4i)T + 4.29e9iT^{2}$$
3 $$1 + (-3.32e7 + 3.32e7i)T - 1.85e15iT^{2}$$
7 $$1 + (3.06e13 + 3.06e13i)T + 1.10e27iT^{2}$$
11 $$1 + 5.53e16T + 2.11e33T^{2}$$
13 $$1 + (-1.61e17 + 1.61e17i)T - 4.42e35iT^{2}$$
17 $$1 + (2.56e19 + 2.56e19i)T + 2.36e39iT^{2}$$
19 $$1 - 4.51e20iT - 8.31e40T^{2}$$
23 $$1 + (-2.16e21 + 2.16e21i)T - 3.76e43iT^{2}$$
29 $$1 - 1.69e23iT - 6.26e46T^{2}$$
31 $$1 - 6.36e23T + 5.29e47T^{2}$$
37 $$1 + (-1.46e24 - 1.46e24i)T + 1.52e50iT^{2}$$
41 $$1 + 4.24e25T + 4.06e51T^{2}$$
43 $$1 + (1.43e26 - 1.43e26i)T - 1.86e52iT^{2}$$
47 $$1 + (-4.39e26 - 4.39e26i)T + 3.21e53iT^{2}$$
53 $$1 + (-1.94e27 + 1.94e27i)T - 1.50e55iT^{2}$$
59 $$1 - 2.69e28iT - 4.64e56T^{2}$$
61 $$1 - 1.09e28T + 1.35e57T^{2}$$
67 $$1 + (-2.17e28 - 2.17e28i)T + 2.71e58iT^{2}$$
71 $$1 - 5.41e29T + 1.73e59T^{2}$$
73 $$1 + (-8.08e28 + 8.08e28i)T - 4.22e59iT^{2}$$
79 $$1 - 1.28e30iT - 5.29e60T^{2}$$
83 $$1 + (3.95e30 - 3.95e30i)T - 2.57e61iT^{2}$$
89 $$1 - 6.87e30iT - 2.40e62T^{2}$$
97 $$1 + (3.25e31 + 3.25e31i)T + 3.77e63iT^{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

−16.58376316301344577216868670428, −13.64293210339163537188431102734, −12.81375837946358549323098676127, −10.67699527948051945052544307754, −9.867597942890053345685478706099, −8.218408962512292950628445355065, −7.03161514748704233929472644640, −3.25040512177457504252007149520, −2.41924351855459688831233944784, −1.16050861565893199520108462473, 0.35797871908249982144063478598, 2.39554685029302205977730674815, 5.14940477876617655534409213101, 6.46778330276938746802644880357, 8.512290310887210164883505416501, 9.206317635078702780272142196517, 10.16492115081353920885099268657, 13.46645820485725397548604766369, 15.35723277808461418867379556920, 15.76766657003408675129090551596