# Properties

 Label 2-50e2-20.19-c0-0-2 Degree $2$ Conductor $2500$ Sign $1$ Analytic cond. $1.24766$ Root an. cond. $1.11698$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s − i·8-s − 9-s − 0.618i·13-s + 16-s − 1.61i·17-s − i·18-s + 0.618·26-s + 1.61·29-s + i·32-s + 1.61·34-s + 36-s − 1.61i·37-s + 0.618·41-s + ⋯
 L(s)  = 1 + i·2-s − 4-s − i·8-s − 9-s − 0.618i·13-s + 16-s − 1.61i·17-s − i·18-s + 0.618·26-s + 1.61·29-s + i·32-s + 1.61·34-s + 36-s − 1.61i·37-s + 0.618·41-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$2500$$    =    $$2^{2} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$1.24766$$ Root analytic conductor: $$1.11698$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2500} (2499, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2500,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8599980639$$ $$L(\frac12)$$ $$\approx$$ $$0.8599980639$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
5 $$1$$
good3 $$1 + T^{2}$$
7 $$1 + T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + 0.618iT - T^{2}$$
17 $$1 + 1.61iT - T^{2}$$
19 $$1 - T^{2}$$
23 $$1 + T^{2}$$
29 $$1 - 1.61T + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + 1.61iT - T^{2}$$
41 $$1 - 0.618T + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + 0.618iT - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - 0.618T + T^{2}$$
67 $$1 + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + 0.618iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 - 1.61T + T^{2}$$
97 $$1 + 1.61iT - 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

−8.966844480668144748648531848841, −8.249922078591038801952009196817, −7.59186397913319923042942574990, −6.81221823988510534100444383719, −6.01201859067437568955349665841, −5.28402419773630207685451165134, −4.65568841957767370800265542642, −3.46840180422705486251243593380, −2.60856274578672415848975273332, −0.62828097749147127048723355992, 1.32037444264783487904565903411, 2.43083820172957114867873953669, 3.27777613987017360413116688629, 4.18288672127242375762140413149, 4.97651031618491700260342894964, 5.94066651665368945124182990021, 6.61735618993872177850138455123, 8.065747707974702558488042021195, 8.377133069776784566232893356696, 9.154101812059798131991724460705