# Properties

 Label 2-50e2-100.71-c0-0-0 Degree $2$ Conductor $2500$ Sign $-0.535 - 0.844i$ Analytic cond. $1.24766$ Root an. cond. $1.11698$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.190 − 0.587i)13-s + (0.309 + 0.951i)16-s + (−1.30 + 0.951i)17-s − 0.999·18-s + 0.618·26-s + (1.30 + 0.951i)29-s − 32-s + (−0.499 − 1.53i)34-s + (0.309 − 0.951i)36-s + (0.5 + 1.53i)37-s + (0.190 + 0.587i)41-s + ⋯
 L(s)  = 1 + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.190 − 0.587i)13-s + (0.309 + 0.951i)16-s + (−1.30 + 0.951i)17-s − 0.999·18-s + 0.618·26-s + (1.30 + 0.951i)29-s − 32-s + (−0.499 − 1.53i)34-s + (0.309 − 0.951i)36-s + (0.5 + 1.53i)37-s + (0.190 + 0.587i)41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2500$$    =    $$2^{2} \cdot 5^{4}$$ Sign: $-0.535 - 0.844i$ Analytic conductor: $$1.24766$$ Root analytic conductor: $$1.11698$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2500} (1751, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2500,\ (\ :0),\ -0.535 - 0.844i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8415761528$$ $$L(\frac12)$$ $$\approx$$ $$0.8415761528$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.309 - 0.951i)T$$
5 $$1$$
good3 $$1 + (-0.309 - 0.951i)T^{2}$$
7 $$1 - T^{2}$$
11 $$1 + (0.809 + 0.587i)T^{2}$$
13 $$1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}$$
17 $$1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}$$
19 $$1 + (-0.309 + 0.951i)T^{2}$$
23 $$1 + (0.809 + 0.587i)T^{2}$$
29 $$1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}$$
31 $$1 + (-0.309 + 0.951i)T^{2}$$
37 $$1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}$$
41 $$1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}$$
43 $$1 - T^{2}$$
47 $$1 + (-0.309 - 0.951i)T^{2}$$
53 $$1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}$$
59 $$1 + (0.809 - 0.587i)T^{2}$$
61 $$1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}$$
67 $$1 + (-0.309 + 0.951i)T^{2}$$
71 $$1 + (-0.309 - 0.951i)T^{2}$$
73 $$1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}$$
79 $$1 + (-0.309 - 0.951i)T^{2}$$
83 $$1 + (-0.309 + 0.951i)T^{2}$$
89 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
97 $$1 + (1.30 + 0.951i)T + (0.309 + 0.951i)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

−9.158547463739272614864621414132, −8.345008201121621420508644503123, −8.005772522897979600576685363862, −6.99778756533648438635666088023, −6.46895173107416869387732625773, −5.51134884825536185985176763639, −4.76990885636083203532849094261, −4.09836404837452125144220482350, −2.66874459260068934211052937482, −1.38158039865059343066219962935, 0.67495917914578661225401303381, 2.08711285937985527778426775329, 2.88275204725785428114534985480, 4.08078958206713210802183495819, 4.45178045845089100633717738064, 5.63238755456889284065269091451, 6.73445601922226179915284642560, 7.31553408366207425601812969308, 8.376465548855396364100881401055, 9.100791510471794249471117039380