# Properties

 Label 2-1859-1.1-c1-0-56 Degree $2$ Conductor $1859$ Sign $1$ Analytic cond. $14.8441$ Root an. cond. $3.85281$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.28·2-s + 1.22·3-s + 3.22·4-s + 2.50·5-s − 2.79·6-s + 4.50·7-s − 2.79·8-s − 1.50·9-s − 5.72·10-s + 11-s + 3.93·12-s − 10.2·14-s + 3.06·15-s − 0.0632·16-s − 3.22·17-s + 3.44·18-s − 2.34·19-s + 8.07·20-s + 5.50·21-s − 2.28·22-s + 5.79·23-s − 3.41·24-s + 1.28·25-s − 5.50·27-s + 14.5·28-s − 0.619·29-s − 7·30-s + ⋯
 L(s)  = 1 − 1.61·2-s + 0.705·3-s + 1.61·4-s + 1.12·5-s − 1.13·6-s + 1.70·7-s − 0.987·8-s − 0.502·9-s − 1.81·10-s + 0.301·11-s + 1.13·12-s − 2.75·14-s + 0.790·15-s − 0.0158·16-s − 0.781·17-s + 0.811·18-s − 0.538·19-s + 1.80·20-s + 1.20·21-s − 0.487·22-s + 1.20·23-s − 0.696·24-s + 0.257·25-s − 1.05·27-s + 2.74·28-s − 0.115·29-s − 1.27·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1859$$    =    $$11 \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$14.8441$$ Root analytic conductor: $$3.85281$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1859,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.505580144$$ $$L(\frac12)$$ $$\approx$$ $$1.505580144$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1 - T$$
13 $$1$$
good2 $$1 + 2.28T + 2T^{2}$$
3 $$1 - 1.22T + 3T^{2}$$
5 $$1 - 2.50T + 5T^{2}$$
7 $$1 - 4.50T + 7T^{2}$$
17 $$1 + 3.22T + 17T^{2}$$
19 $$1 + 2.34T + 19T^{2}$$
23 $$1 - 5.79T + 23T^{2}$$
29 $$1 + 0.619T + 29T^{2}$$
31 $$1 - 6.01T + 31T^{2}$$
37 $$1 - 5.06T + 37T^{2}$$
41 $$1 + 8.36T + 41T^{2}$$
43 $$1 - 11.0T + 43T^{2}$$
47 $$1 - 6.74T + 47T^{2}$$
53 $$1 + 4.06T + 53T^{2}$$
59 $$1 - 11.0T + 59T^{2}$$
61 $$1 + 0.144T + 61T^{2}$$
67 $$1 - 8.72T + 67T^{2}$$
71 $$1 + 1.71T + 71T^{2}$$
73 $$1 + 2T + 73T^{2}$$
79 $$1 - 1.87T + 79T^{2}$$
83 $$1 + 5.38T + 83T^{2}$$
89 $$1 + 8.50T + 89T^{2}$$
97 $$1 - 0.112T + 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

−9.009569834858445862454723873272, −8.634420502573404156460062679816, −8.031413881648961474263964814963, −7.23331895248129230602853147070, −6.31453507395188991606358542423, −5.32272985522975071608126966034, −4.31797478588735510022180079166, −2.57921639314973295547991621131, −2.05737043556972920182119728839, −1.10232249550463505653129708393, 1.10232249550463505653129708393, 2.05737043556972920182119728839, 2.57921639314973295547991621131, 4.31797478588735510022180079166, 5.32272985522975071608126966034, 6.31453507395188991606358542423, 7.23331895248129230602853147070, 8.031413881648961474263964814963, 8.634420502573404156460062679816, 9.009569834858445862454723873272