# Properties

 Degree $2$ Conductor $112$ Sign $0.895 + 0.444i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)3-s + (1.5 + 0.866i)5-s + (2 − 1.73i)7-s + (1 − 1.73i)9-s + (−1.5 + 0.866i)11-s − 1.73i·15-s + (−4.5 + 2.59i)17-s + (−3.5 + 6.06i)19-s + (−2.5 − 0.866i)21-s + (7.5 + 4.33i)23-s + (−1 − 1.73i)25-s − 5·27-s − 6·29-s + (−2.5 − 4.33i)31-s + (1.5 + 0.866i)33-s + ⋯
 L(s)  = 1 + (−0.288 − 0.499i)3-s + (0.670 + 0.387i)5-s + (0.755 − 0.654i)7-s + (0.333 − 0.577i)9-s + (−0.452 + 0.261i)11-s − 0.447i·15-s + (−1.09 + 0.630i)17-s + (−0.802 + 1.39i)19-s + (−0.545 − 0.188i)21-s + (1.56 + 0.902i)23-s + (−0.200 − 0.346i)25-s − 0.962·27-s − 1.11·29-s + (−0.449 − 0.777i)31-s + (0.261 + 0.150i)33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$112$$    =    $$2^{4} \cdot 7$$ Sign: $0.895 + 0.444i$ Motivic weight: $$1$$ Character: $\chi_{112} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 112,\ (\ :1/2),\ 0.895 + 0.444i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.05190 - 0.246421i$$ $$L(\frac12)$$ $$\approx$$ $$1.05190 - 0.246421i$$ $$L(\frac{3}{2})$$ 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 + (-2 + 1.73i)T$$
good3 $$1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-7.5 - 4.33i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 6.92iT - 41T^{2}$$
43 $$1 + 3.46iT - 43T^{2}$$
47 $$1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-4.5 + 2.59i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 - 3.46iT - 71T^{2}$$
73 $$1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (4.5 + 2.59i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 - 12T + 83T^{2}$$
89 $$1 + (10.5 + 6.06i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 - 6.92iT - 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

−13.36435495900951915893436555353, −12.74792982753816727685264134953, −11.35668188794049704482501219594, −10.53881676871788679563498254985, −9.391354172952845164126764313371, −7.894237687879887047672717842079, −6.85205033185553107187980022870, −5.74024338411001776721315514030, −4.07424526339612746354847297653, −1.82698880999477658675413412774, 2.29797226428550356109097376842, 4.71525060893812083005003990248, 5.35765362414082962117536760423, 6.99159975970254225762761686382, 8.572229715794336102069115036051, 9.343825111569200105677304811287, 10.77906089588489236258136790061, 11.29943998937957979654265389195, 12.90991557214120478155665202877, 13.48390054678584073455726518985