# Properties

 Label 2-3040-1.1-c1-0-22 Degree $2$ Conductor $3040$ Sign $1$ Analytic cond. $24.2745$ Root an. cond. $4.92691$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.56·3-s − 5-s − 4.56·7-s + 3.56·9-s + 4·11-s + 2.56·13-s − 2.56·15-s − 0.561·17-s + 19-s − 11.6·21-s + 5.68·23-s + 25-s + 1.43·27-s − 3.43·29-s + 5.12·31-s + 10.2·33-s + 4.56·35-s − 1.12·37-s + 6.56·39-s + 8.24·41-s + 2·43-s − 3.56·45-s − 3.12·47-s + 13.8·49-s − 1.43·51-s + 6.56·53-s − 4·55-s + ⋯
 L(s)  = 1 + 1.47·3-s − 0.447·5-s − 1.72·7-s + 1.18·9-s + 1.20·11-s + 0.710·13-s − 0.661·15-s − 0.136·17-s + 0.229·19-s − 2.54·21-s + 1.18·23-s + 0.200·25-s + 0.276·27-s − 0.638·29-s + 0.920·31-s + 1.78·33-s + 0.771·35-s − 0.184·37-s + 1.05·39-s + 1.28·41-s + 0.304·43-s − 0.530·45-s − 0.455·47-s + 1.97·49-s − 0.201·51-s + 0.901·53-s − 0.539·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3040$$    =    $$2^{5} \cdot 5 \cdot 19$$ Sign: $1$ Analytic conductor: $$24.2745$$ Root analytic conductor: $$4.92691$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 3040,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.678057748$$ $$L(\frac12)$$ $$\approx$$ $$2.678057748$$ $$L(\frac{3}{2})$$ 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 + T$$
19 $$1 - T$$
good3 $$1 - 2.56T + 3T^{2}$$
7 $$1 + 4.56T + 7T^{2}$$
11 $$1 - 4T + 11T^{2}$$
13 $$1 - 2.56T + 13T^{2}$$
17 $$1 + 0.561T + 17T^{2}$$
23 $$1 - 5.68T + 23T^{2}$$
29 $$1 + 3.43T + 29T^{2}$$
31 $$1 - 5.12T + 31T^{2}$$
37 $$1 + 1.12T + 37T^{2}$$
41 $$1 - 8.24T + 41T^{2}$$
43 $$1 - 2T + 43T^{2}$$
47 $$1 + 3.12T + 47T^{2}$$
53 $$1 - 6.56T + 53T^{2}$$
59 $$1 + 12.8T + 59T^{2}$$
61 $$1 + 4.87T + 61T^{2}$$
67 $$1 - 10.5T + 67T^{2}$$
71 $$1 - 8T + 71T^{2}$$
73 $$1 - 14.8T + 73T^{2}$$
79 $$1 + 5.12T + 79T^{2}$$
83 $$1 - 14T + 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 1.12T + 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

−8.895318358094823467828782393939, −8.094466549504147361129515408084, −7.22262196112032373566630806190, −6.63916936997000433512436559977, −5.91786885671210340845076591957, −4.46522482451246616473928113876, −3.54985649169236479077483425639, −3.34793456630333448937126847499, −2.35582090438890933964225614140, −0.948917446234411268788742012051, 0.948917446234411268788742012051, 2.35582090438890933964225614140, 3.34793456630333448937126847499, 3.54985649169236479077483425639, 4.46522482451246616473928113876, 5.91786885671210340845076591957, 6.63916936997000433512436559977, 7.22262196112032373566630806190, 8.094466549504147361129515408084, 8.895318358094823467828782393939