# Properties

 Label 2-1792-16.5-c1-0-0 Degree $2$ Conductor $1792$ Sign $-0.991 + 0.130i$ Analytic cond. $14.3091$ Root an. cond. $3.78274$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.328 + 0.328i)3-s + (−1.40 − 1.40i)5-s + i·7-s + 2.78i·9-s + (−0.444 − 0.444i)11-s + (−2.25 + 2.25i)13-s + 0.919·15-s + 4.85·17-s + (0.114 − 0.114i)19-s + (−0.328 − 0.328i)21-s − 3.20i·23-s − 1.06i·25-s + (−1.89 − 1.89i)27-s + (0.997 − 0.997i)29-s − 5.34·31-s + ⋯
 L(s)  = 1 + (−0.189 + 0.189i)3-s + (−0.626 − 0.626i)5-s + 0.377i·7-s + 0.928i·9-s + (−0.133 − 0.133i)11-s + (−0.625 + 0.625i)13-s + 0.237·15-s + 1.17·17-s + (0.0261 − 0.0261i)19-s + (−0.0715 − 0.0715i)21-s − 0.668i·23-s − 0.213i·25-s + (−0.365 − 0.365i)27-s + (0.185 − 0.185i)29-s − 0.959·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1792$$    =    $$2^{8} \cdot 7$$ Sign: $-0.991 + 0.130i$ Analytic conductor: $$14.3091$$ Root analytic conductor: $$3.78274$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1792} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1792,\ (\ :1/2),\ -0.991 + 0.130i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.06823327470$$ $$L(\frac12)$$ $$\approx$$ $$0.06823327470$$ $$L(\frac{3}{2})$$ 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 - iT$$
good3 $$1 + (0.328 - 0.328i)T - 3iT^{2}$$
5 $$1 + (1.40 + 1.40i)T + 5iT^{2}$$
11 $$1 + (0.444 + 0.444i)T + 11iT^{2}$$
13 $$1 + (2.25 - 2.25i)T - 13iT^{2}$$
17 $$1 - 4.85T + 17T^{2}$$
19 $$1 + (-0.114 + 0.114i)T - 19iT^{2}$$
23 $$1 + 3.20iT - 23T^{2}$$
29 $$1 + (-0.997 + 0.997i)T - 29iT^{2}$$
31 $$1 + 5.34T + 31T^{2}$$
37 $$1 + (2.03 + 2.03i)T + 37iT^{2}$$
41 $$1 - 9.57iT - 41T^{2}$$
43 $$1 + (6.86 + 6.86i)T + 43iT^{2}$$
47 $$1 + 9.70T + 47T^{2}$$
53 $$1 + (7.64 + 7.64i)T + 53iT^{2}$$
59 $$1 + (1.50 + 1.50i)T + 59iT^{2}$$
61 $$1 + (-1.74 + 1.74i)T - 61iT^{2}$$
67 $$1 + (8.96 - 8.96i)T - 67iT^{2}$$
71 $$1 - 7.18iT - 71T^{2}$$
73 $$1 - 9.04iT - 73T^{2}$$
79 $$1 + 9.58T + 79T^{2}$$
83 $$1 + (5.30 - 5.30i)T - 83iT^{2}$$
89 $$1 + 2.49iT - 89T^{2}$$
97 $$1 - 5.89T + 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

−9.807803067337186499762822633628, −8.743533594788682317295133852259, −8.162991700102944933195426140413, −7.49585079613164579331264688607, −6.50947284306868845709281364420, −5.39073266650766868063886315145, −4.88952895000227328107802569244, −4.02188444450164919367136819931, −2.85115824314597269904519172205, −1.66376964208160175781251552295, 0.02622507490562297477307873735, 1.48058813756694946884888447314, 3.19131641209381131830169518769, 3.48766533754416428539297250488, 4.77630336304281912733003851473, 5.69637310254353535167033081294, 6.56848971792303948096564816426, 7.50109858034834384136502526003, 7.67559298909664268739799517529, 8.916021840185136058016698651474