# Properties

 Label 2-40e2-1.1-c3-0-28 Degree $2$ Conductor $1600$ Sign $1$ Analytic cond. $94.4030$ Root an. cond. $9.71612$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 27·9-s + 92·13-s + 104·17-s − 130·29-s − 396·37-s + 230·41-s − 343·49-s − 572·53-s + 830·61-s + 592·73-s + 729·81-s + 1.67e3·89-s + 1.81e3·97-s − 598·101-s + 1.74e3·109-s + 1.32e3·113-s − 2.48e3·117-s + ⋯
 L(s)  = 1 − 9-s + 1.96·13-s + 1.48·17-s − 0.832·29-s − 1.75·37-s + 0.876·41-s − 49-s − 1.48·53-s + 1.74·61-s + 0.949·73-s + 81-s + 1.98·89-s + 1.90·97-s − 0.589·101-s + 1.53·109-s + 1.10·113-s − 1.96·117-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$94.4030$$ Root analytic conductor: $$9.71612$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1600,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.170747332$$ $$L(\frac12)$$ $$\approx$$ $$2.170747332$$ $$L(\frac{5}{2})$$ 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$$
good3 $$1 + p^{3} T^{2}$$
7 $$1 + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 - 92 T + p^{3} T^{2}$$
17 $$1 - 104 T + p^{3} T^{2}$$
19 $$1 + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + 130 T + p^{3} T^{2}$$
31 $$1 + p^{3} T^{2}$$
37 $$1 + 396 T + p^{3} T^{2}$$
41 $$1 - 230 T + p^{3} T^{2}$$
43 $$1 + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + 572 T + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 - 830 T + p^{3} T^{2}$$
67 $$1 + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 - 592 T + p^{3} T^{2}$$
79 $$1 + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 - 1670 T + p^{3} T^{2}$$
97 $$1 - 1816 T + p^{3} T^{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

−8.910638815477927300418543725229, −8.322731025003355346338596659987, −7.60334988348691863282311097380, −6.43705211097272875357808887672, −5.82621934776736941704139075647, −5.09289003427586174508546352719, −3.67128040244791292349057637806, −3.26210713818336384030711292775, −1.82182526404571793481162085201, −0.72776802490996063526525155967, 0.72776802490996063526525155967, 1.82182526404571793481162085201, 3.26210713818336384030711292775, 3.67128040244791292349057637806, 5.09289003427586174508546352719, 5.82621934776736941704139075647, 6.43705211097272875357808887672, 7.60334988348691863282311097380, 8.322731025003355346338596659987, 8.910638815477927300418543725229