# Properties

 Label 2-40e2-40.29-c1-0-9 Degree $2$ Conductor $1600$ Sign $0.663 - 0.748i$ Analytic cond. $12.7760$ Root an. cond. $3.57436$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.73·3-s + 4.73i·7-s + 4.46·9-s − 3.46i·11-s + 3.46·13-s − 3.46i·17-s − 2i·19-s − 12.9i·21-s + 2.19i·23-s − 3.99·27-s − 2.53·31-s + 9.46i·33-s − 6·37-s − 9.46·39-s + 9.46·41-s + ⋯
 L(s)  = 1 − 1.57·3-s + 1.78i·7-s + 1.48·9-s − 1.04i·11-s + 0.960·13-s − 0.840i·17-s − 0.458i·19-s − 2.82i·21-s + 0.457i·23-s − 0.769·27-s − 0.455·31-s + 1.64i·33-s − 0.986·37-s − 1.51·39-s + 1.47·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $0.663 - 0.748i$ Analytic conductor: $$12.7760$$ Root analytic conductor: $$3.57436$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1600} (1249, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1600,\ (\ :1/2),\ 0.663 - 0.748i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9064526098$$ $$L(\frac12)$$ $$\approx$$ $$0.9064526098$$ $$L(\frac{3}{2})$$ 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 + 2.73T + 3T^{2}$$
7 $$1 - 4.73iT - 7T^{2}$$
11 $$1 + 3.46iT - 11T^{2}$$
13 $$1 - 3.46T + 13T^{2}$$
17 $$1 + 3.46iT - 17T^{2}$$
19 $$1 + 2iT - 19T^{2}$$
23 $$1 - 2.19iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 2.53T + 31T^{2}$$
37 $$1 + 6T + 37T^{2}$$
41 $$1 - 9.46T + 41T^{2}$$
43 $$1 - 0.196T + 43T^{2}$$
47 $$1 - 2.19iT - 47T^{2}$$
53 $$1 - 10.3T + 53T^{2}$$
59 $$1 - 6iT - 59T^{2}$$
61 $$1 + 0.928iT - 61T^{2}$$
67 $$1 + 0.196T + 67T^{2}$$
71 $$1 - 16.3T + 71T^{2}$$
73 $$1 + 6.39iT - 73T^{2}$$
79 $$1 - 12T + 79T^{2}$$
83 $$1 + 1.26T + 83T^{2}$$
89 $$1 - 12.9T + 89T^{2}$$
97 $$1 - 14.3iT - 97T^{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

−9.341346696471835244945763372174, −8.930876048124846572319040778156, −7.952299432680398701348327371362, −6.74472405175734288215549941783, −6.05629870827409417482628573842, −5.52903270288770044722426606934, −4.97768922843258220212581866855, −3.58631853016481276828094017169, −2.39130949933524008626340830293, −0.870757098151957887564925037967, 0.63697300345061816541401076135, 1.66948858401971837384382269558, 3.76903046843412352088818617593, 4.26845184809000422698765713610, 5.19569184620023023857101114300, 6.13545737207515525797784285937, 6.81879842586343325721072894057, 7.39728205436452180731208985992, 8.357782858638802070712697278761, 9.680857412858175445327760307917