# Properties

 Degree $2$ Conductor $1568$ Sign $0.968 - 0.250i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.618 + 1.07i)3-s + (1.61 − 2.80i)5-s + (0.736 − 1.27i)9-s + (3.23 + 5.60i)11-s − 0.763·13-s + 4·15-s + (2.23 + 3.87i)17-s + (−0.618 + 1.07i)19-s + (−2 + 3.46i)23-s + (−2.73 − 4.73i)25-s + 5.52·27-s − 4.47·29-s + (−1.23 − 2.14i)31-s + (−3.99 + 6.92i)33-s + (2.23 − 3.87i)37-s + ⋯
 L(s)  = 1 + (0.356 + 0.618i)3-s + (0.723 − 1.25i)5-s + (0.245 − 0.424i)9-s + (0.975 + 1.68i)11-s − 0.211·13-s + 1.03·15-s + (0.542 + 0.939i)17-s + (−0.141 + 0.245i)19-s + (−0.417 + 0.722i)23-s + (−0.547 − 0.947i)25-s + 1.06·27-s − 0.830·29-s + (−0.222 − 0.384i)31-s + (−0.696 + 1.20i)33-s + (0.367 − 0.636i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1568$$    =    $$2^{5} \cdot 7^{2}$$ Sign: $0.968 - 0.250i$ Motivic weight: $$1$$ Character: $\chi_{1568} (961, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1568,\ (\ :1/2),\ 0.968 - 0.250i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.430530251$$ $$L(\frac12)$$ $$\approx$$ $$2.430530251$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 + (-0.618 - 1.07i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-1.61 + 2.80i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-3.23 - 5.60i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + 0.763T + 13T^{2}$$
17 $$1 + (-2.23 - 3.87i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (0.618 - 1.07i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 4.47T + 29T^{2}$$
31 $$1 + (1.23 + 2.14i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-2.23 + 3.87i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 8.47T + 41T^{2}$$
43 $$1 - 6.47T + 43T^{2}$$
47 $$1 + (-5.23 + 9.06i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (4.61 + 7.99i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-5.61 + 9.73i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 4.94T + 71T^{2}$$
73 $$1 + (1.47 + 2.54i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (6.47 - 11.2i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 9.23T + 83T^{2}$$
89 $$1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 12.4T + 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

−9.428223225127409840227544167436, −9.072059975504621395228337445566, −7.981649269363956728136887973432, −7.11196009201750804713482240043, −6.04699202568584551542536949452, −5.27750310080199303799399355696, −4.21340540097511975254601781891, −3.91358325363214017269895142489, −2.14106352375399772421463332286, −1.27834936501295305893782705802, 1.12119898595768259774953155189, 2.47452599232644446512216152154, 3.01130244317053824634602620685, 4.20086765688749714920559095768, 5.63322644018677180133594010184, 6.21096437715703021896409158587, 7.03212659754555051377460567314, 7.61859524478455111339523926514, 8.624726065687283472881688273242, 9.337668940882816269874739626295