# Properties

 Label 2-418-1.1-c1-0-8 Degree $2$ Conductor $418$ Sign $1$ Analytic cond. $3.33774$ Root an. cond. $1.82695$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 2.39·3-s + 4-s − 0.391·5-s + 2.39·6-s − 1.71·7-s + 8-s + 2.71·9-s − 0.391·10-s − 11-s + 2.39·12-s + 3.71·13-s − 1.71·14-s − 0.935·15-s + 16-s + 5.43·17-s + 2.71·18-s − 19-s − 0.391·20-s − 4.11·21-s − 22-s − 3.43·23-s + 2.39·24-s − 4.84·25-s + 3.71·26-s − 0.672·27-s − 1.71·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.38·3-s + 0.5·4-s − 0.175·5-s + 0.976·6-s − 0.649·7-s + 0.353·8-s + 0.906·9-s − 0.123·10-s − 0.301·11-s + 0.690·12-s + 1.03·13-s − 0.459·14-s − 0.241·15-s + 0.250·16-s + 1.31·17-s + 0.640·18-s − 0.229·19-s − 0.0875·20-s − 0.896·21-s − 0.213·22-s − 0.716·23-s + 0.488·24-s − 0.969·25-s + 0.729·26-s − 0.129·27-s − 0.324·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$418$$    =    $$2 \cdot 11 \cdot 19$$ Sign: $1$ Analytic conductor: $$3.33774$$ Root analytic conductor: $$1.82695$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 418,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.896988465$$ $$L(\frac12)$$ $$\approx$$ $$2.896988465$$ $$L(\frac{3}{2})$$ 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$$
11 $$1 + T$$
19 $$1 + T$$
good3 $$1 - 2.39T + 3T^{2}$$
5 $$1 + 0.391T + 5T^{2}$$
7 $$1 + 1.71T + 7T^{2}$$
13 $$1 - 3.71T + 13T^{2}$$
17 $$1 - 5.43T + 17T^{2}$$
23 $$1 + 3.43T + 23T^{2}$$
29 $$1 + 3.82T + 29T^{2}$$
31 $$1 + 3.06T + 31T^{2}$$
37 $$1 + 4.65T + 37T^{2}$$
41 $$1 - 1.06T + 41T^{2}$$
43 $$1 + 3.73T + 43T^{2}$$
47 $$1 + 8.22T + 47T^{2}$$
53 $$1 + 1.21T + 53T^{2}$$
59 $$1 - 12.4T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 + 1.71T + 67T^{2}$$
71 $$1 - 1.04T + 71T^{2}$$
73 $$1 - 13.6T + 73T^{2}$$
79 $$1 - 13.0T + 79T^{2}$$
83 $$1 - 8.51T + 83T^{2}$$
89 $$1 - 17.6T + 89T^{2}$$
97 $$1 + 4.65T + 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

−11.31468820881441681215976561818, −10.16050325440825332556907548324, −9.400503517462249451900216195901, −8.275157492226730483224248510818, −7.69558298026064181626568328514, −6.46899371741935064110887626858, −5.41305306052668382208684587190, −3.75176066846497908511832953172, −3.38782446174387078263704721746, −1.99354725019068991302511362310, 1.99354725019068991302511362310, 3.38782446174387078263704721746, 3.75176066846497908511832953172, 5.41305306052668382208684587190, 6.46899371741935064110887626858, 7.69558298026064181626568328514, 8.275157492226730483224248510818, 9.400503517462249451900216195901, 10.16050325440825332556907548324, 11.31468820881441681215976561818