# Properties

 Label 2-1840-5.4-c1-0-3 Degree $2$ Conductor $1840$ Sign $-0.507 + 0.861i$ Analytic cond. $14.6924$ Root an. cond. $3.83307$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.51i·3-s + (−1.92 − 1.13i)5-s + 4.64i·7-s − 3.32·9-s + 1.64·11-s − 1.91i·13-s + (2.85 − 4.84i)15-s + 0.969i·17-s − 4.91·19-s − 11.6·21-s + i·23-s + (2.42 + 4.37i)25-s − 0.825i·27-s − 7.48·29-s − 7.77·31-s + ⋯
 L(s)  = 1 + 1.45i·3-s + (−0.861 − 0.507i)5-s + 1.75i·7-s − 1.10·9-s + 0.497·11-s − 0.531i·13-s + (0.737 − 1.25i)15-s + 0.235i·17-s − 1.12·19-s − 2.54·21-s + 0.208i·23-s + (0.484 + 0.874i)25-s − 0.158i·27-s − 1.38·29-s − 1.39·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1840$$    =    $$2^{4} \cdot 5 \cdot 23$$ Sign: $-0.507 + 0.861i$ Analytic conductor: $$14.6924$$ Root analytic conductor: $$3.83307$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1840} (369, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1840,\ (\ :1/2),\ -0.507 + 0.861i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4968896719$$ $$L(\frac12)$$ $$\approx$$ $$0.4968896719$$ $$L(\frac{3}{2})$$ 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 + (1.92 + 1.13i)T$$
23 $$1 - iT$$
good3 $$1 - 2.51iT - 3T^{2}$$
7 $$1 - 4.64iT - 7T^{2}$$
11 $$1 - 1.64T + 11T^{2}$$
13 $$1 + 1.91iT - 13T^{2}$$
17 $$1 - 0.969iT - 17T^{2}$$
19 $$1 + 4.91T + 19T^{2}$$
29 $$1 + 7.48T + 29T^{2}$$
31 $$1 + 7.77T + 31T^{2}$$
37 $$1 - 7.63iT - 37T^{2}$$
41 $$1 - 5.79T + 41T^{2}$$
43 $$1 + 3.77iT - 43T^{2}$$
47 $$1 - 2.22iT - 47T^{2}$$
53 $$1 + 11.2iT - 53T^{2}$$
59 $$1 - 10.1T + 59T^{2}$$
61 $$1 - 6.06T + 61T^{2}$$
67 $$1 + 12.0iT - 67T^{2}$$
71 $$1 + 3.01T + 71T^{2}$$
73 $$1 + 16.6iT - 73T^{2}$$
79 $$1 + 0.163T + 79T^{2}$$
83 $$1 + 5.66iT - 83T^{2}$$
89 $$1 - 3.90T + 89T^{2}$$
97 $$1 - 1.79iT - 97T^{2}$$
show less
$$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

−9.527112899346983569932053485264, −9.007527408416018541495190192978, −8.557674356238466238511845656502, −7.68781141756687540576361515545, −6.34918073758110925545117192255, −5.43738663000797774391253721950, −4.95674362587552535375056111075, −3.93267034635741324815670278759, −3.34266691492945401110820064807, −2.03886944756705237537917863312, 0.19017355555301537217612454725, 1.29547439711868513604474785249, 2.44365502649084994924220033570, 3.92063541066802323555310319936, 4.13712575539966498016506387436, 5.79011515598005953058453717793, 6.91520441828645483109253338823, 7.00718098005757999512078545475, 7.64869007376296826119311054774, 8.411523879427727337537170225133