# Properties

 Label 2-2850-1.1-c1-0-18 Degree $2$ Conductor $2850$ Sign $1$ Analytic cond. $22.7573$ Root an. cond. $4.77046$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s − 6-s + 1.07·7-s + 8-s + 9-s + 6.34·11-s − 12-s − 3.41·13-s + 1.07·14-s + 16-s + 5.41·17-s + 18-s + 19-s − 1.07·21-s + 6.34·22-s − 6.34·23-s − 24-s − 3.41·26-s − 27-s + 1.07·28-s − 0.340·29-s + 1.07·31-s + 32-s − 6.34·33-s + 5.41·34-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.407·7-s + 0.353·8-s + 0.333·9-s + 1.91·11-s − 0.288·12-s − 0.948·13-s + 0.288·14-s + 0.250·16-s + 1.31·17-s + 0.235·18-s + 0.229·19-s − 0.235·21-s + 1.35·22-s − 1.32·23-s − 0.204·24-s − 0.670·26-s − 0.192·27-s + 0.203·28-s − 0.0631·29-s + 0.193·31-s + 0.176·32-s − 1.10·33-s + 0.929·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2850$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 19$$ Sign: $1$ Analytic conductor: $$22.7573$$ Root analytic conductor: $$4.77046$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2850} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2850,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.864863645$$ $$L(\frac12)$$ $$\approx$$ $$2.864863645$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 + T$$
5 $$1$$
19 $$1 - T$$
good7 $$1 - 1.07T + 7T^{2}$$
11 $$1 - 6.34T + 11T^{2}$$
13 $$1 + 3.41T + 13T^{2}$$
17 $$1 - 5.41T + 17T^{2}$$
23 $$1 + 6.34T + 23T^{2}$$
29 $$1 + 0.340T + 29T^{2}$$
31 $$1 - 1.07T + 31T^{2}$$
37 $$1 + 3.41T + 37T^{2}$$
41 $$1 - 7.60T + 41T^{2}$$
43 $$1 - 11.1T + 43T^{2}$$
47 $$1 - 6.34T + 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 - 0.738T + 59T^{2}$$
61 $$1 + 2.68T + 61T^{2}$$
67 $$1 + 2.83T + 67T^{2}$$
71 $$1 + 2.83T + 71T^{2}$$
73 $$1 + 6.83T + 73T^{2}$$
79 $$1 + 1.07T + 79T^{2}$$
83 $$1 + 0.894T + 83T^{2}$$
89 $$1 - 6.92T + 89T^{2}$$
97 $$1 - 3.65T + 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

−8.855125319018892942729613298227, −7.66336827210750210261875024648, −7.28941650460627868143686261376, −6.17861540134138408290630075127, −5.86342731974160431804891755937, −4.79888333972843455806985589619, −4.17544800644602202315273018135, −3.35801230543915315290564186352, −2.04476563802979756023364690474, −1.03984464621972180411597826029, 1.03984464621972180411597826029, 2.04476563802979756023364690474, 3.35801230543915315290564186352, 4.17544800644602202315273018135, 4.79888333972843455806985589619, 5.86342731974160431804891755937, 6.17861540134138408290630075127, 7.28941650460627868143686261376, 7.66336827210750210261875024648, 8.855125319018892942729613298227