# Properties

 Label 2-1340-1340.659-c0-0-0 Degree $2$ Conductor $1340$ Sign $0.998 + 0.0490i$ Analytic cond. $0.668747$ Root an. cond. $0.817769$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.327 − 0.945i)2-s + (−0.653 − 1.43i)3-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (−1.13 + 1.08i)6-s + (−1.95 − 0.376i)7-s + (0.841 + 0.540i)8-s + (−0.963 + 1.11i)9-s + (0.580 + 0.814i)10-s + (1.39 + 0.720i)12-s + (0.283 + 1.97i)14-s + (1.02 + 1.18i)15-s + (0.235 − 0.971i)16-s + (1.36 + 0.546i)18-s + (0.580 − 0.814i)20-s + (0.737 + 3.04i)21-s + ⋯
 L(s)  = 1 + (−0.327 − 0.945i)2-s + (−0.653 − 1.43i)3-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (−1.13 + 1.08i)6-s + (−1.95 − 0.376i)7-s + (0.841 + 0.540i)8-s + (−0.963 + 1.11i)9-s + (0.580 + 0.814i)10-s + (1.39 + 0.720i)12-s + (0.283 + 1.97i)14-s + (1.02 + 1.18i)15-s + (0.235 − 0.971i)16-s + (1.36 + 0.546i)18-s + (0.580 − 0.814i)20-s + (0.737 + 3.04i)21-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1340$$    =    $$2^{2} \cdot 5 \cdot 67$$ Sign: $0.998 + 0.0490i$ Analytic conductor: $$0.668747$$ Root analytic conductor: $$0.817769$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1340} (659, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1340,\ (\ :0),\ 0.998 + 0.0490i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1377745596$$ $$L(\frac12)$$ $$\approx$$ $$0.1377745596$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.327 + 0.945i)T$$
5 $$1 + (0.959 - 0.281i)T$$
67 $$1 + (-0.928 - 0.371i)T$$
good3 $$1 + (0.653 + 1.43i)T + (-0.654 + 0.755i)T^{2}$$
7 $$1 + (1.95 + 0.376i)T + (0.928 + 0.371i)T^{2}$$
11 $$1 + (-0.0475 - 0.998i)T^{2}$$
13 $$1 + (0.995 - 0.0950i)T^{2}$$
17 $$1 + (-0.235 - 0.971i)T^{2}$$
19 $$1 + (-0.928 + 0.371i)T^{2}$$
23 $$1 + (-0.995 - 0.0950i)T + (0.981 + 0.189i)T^{2}$$
29 $$1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 + (0.995 + 0.0950i)T^{2}$$
37 $$1 + (0.5 + 0.866i)T^{2}$$
41 $$1 + (-0.0883 + 0.0353i)T + (0.723 - 0.690i)T^{2}$$
43 $$1 + (0.264 - 1.83i)T + (-0.959 - 0.281i)T^{2}$$
47 $$1 + (1.03 - 1.44i)T + (-0.327 - 0.945i)T^{2}$$
53 $$1 + (0.959 - 0.281i)T^{2}$$
59 $$1 + (-0.415 - 0.909i)T^{2}$$
61 $$1 + (1.38 - 1.32i)T + (0.0475 - 0.998i)T^{2}$$
71 $$1 + (-0.235 + 0.971i)T^{2}$$
73 $$1 + (-0.0475 + 0.998i)T^{2}$$
79 $$1 + (-0.580 - 0.814i)T^{2}$$
83 $$1 + (-0.462 + 1.90i)T + (-0.888 - 0.458i)T^{2}$$
89 $$1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2}$$
97 $$1 + (0.5 + 0.866i)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

−9.870218796120561462538956603808, −9.148635914178771256733006237763, −8.002949254904411352968679868717, −7.38606005802106352678210454287, −6.70236039537464077944419039683, −5.94320079633574648342366671223, −4.45987914096870670961000381854, −3.37379193052877670450676378542, −2.67896435307456434736608912155, −1.03652046509493954394949484314, 0.17551713050757869359012690629, 3.34985294253609396317227837812, 3.85566648335290344004003635723, 4.97744813466369644834039999737, 5.53801953140480117283424215960, 6.57110066048506986023398814264, 7.13282040866237937191876764618, 8.452544443047057580657254683581, 9.166118042712494610820021244529, 9.593658782464571322590117522348