# Properties

 Label 2-1110-1.1-c1-0-21 Degree $2$ Conductor $1110$ Sign $-1$ Analytic cond. $8.86339$ Root an. cond. $2.97714$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 6.12·11-s − 12-s + 5.68·13-s − 14-s + 15-s + 16-s − 5·17-s + 18-s + 0.561·19-s − 20-s + 21-s − 6.12·22-s − 6.56·23-s − 24-s + 25-s + 5.68·26-s − 27-s − 28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.84·11-s − 0.288·12-s + 1.57·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s + 0.128·19-s − 0.223·20-s + 0.218·21-s − 1.30·22-s − 1.36·23-s − 0.204·24-s + 0.200·25-s + 1.11·26-s − 0.192·27-s − 0.188·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1110$$    =    $$2 \cdot 3 \cdot 5 \cdot 37$$ Sign: $-1$ Analytic conductor: $$8.86339$$ Root analytic conductor: $$2.97714$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1110} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1110,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 + T$$
5 $$1 + T$$
37 $$1 - T$$
good7 $$1 + T + 7T^{2}$$
11 $$1 + 6.12T + 11T^{2}$$
13 $$1 - 5.68T + 13T^{2}$$
17 $$1 + 5T + 17T^{2}$$
19 $$1 - 0.561T + 19T^{2}$$
23 $$1 + 6.56T + 23T^{2}$$
29 $$1 + 6.68T + 29T^{2}$$
31 $$1 + 3.56T + 31T^{2}$$
41 $$1 - 8.68T + 41T^{2}$$
43 $$1 + 12.6T + 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 - 5.24T + 53T^{2}$$
59 $$1 - 4.24T + 59T^{2}$$
61 $$1 + 13.8T + 61T^{2}$$
67 $$1 - 13.1T + 67T^{2}$$
71 $$1 - 1.12T + 71T^{2}$$
73 $$1 + 6.56T + 73T^{2}$$
79 $$1 - 14.2T + 79T^{2}$$
83 $$1 + 7.43T + 83T^{2}$$
89 $$1 - 5.68T + 89T^{2}$$
97 $$1 - 3.31T + 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

−9.600516873136888235247570747972, −8.369103931345263489013047968798, −7.73589678570944847890062228996, −6.68809362990095016628647885460, −5.93895383136299238805627461161, −5.17864061466260350396003325547, −4.16743143517120259529648606362, −3.30027531686472465337447397895, −1.99602686798888578561163090851, 0, 1.99602686798888578561163090851, 3.30027531686472465337447397895, 4.16743143517120259529648606362, 5.17864061466260350396003325547, 5.93895383136299238805627461161, 6.68809362990095016628647885460, 7.73589678570944847890062228996, 8.369103931345263489013047968798, 9.600516873136888235247570747972