# Properties

 Label 2-5520-92.91-c1-0-39 Degree $2$ Conductor $5520$ Sign $-0.310 - 0.950i$ Analytic cond. $44.0774$ Root an. cond. $6.63908$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + i·3-s + i·5-s + 1.70·7-s − 9-s + 3.79·11-s + 1.61·13-s − 15-s + 2.05i·17-s − 0.518·19-s + 1.70i·21-s + (−4.69 − 0.988i)23-s − 25-s − i·27-s − 2.88·29-s + 4.16i·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.447i·5-s + 0.643·7-s − 0.333·9-s + 1.14·11-s + 0.448·13-s − 0.258·15-s + 0.498i·17-s − 0.119·19-s + 0.371i·21-s + (−0.978 − 0.206i)23-s − 0.200·25-s − 0.192i·27-s − 0.535·29-s + 0.748i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5520$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 23$$ Sign: $-0.310 - 0.950i$ Analytic conductor: $$44.0774$$ Root analytic conductor: $$6.63908$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{5520} (1471, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5520,\ (\ :1/2),\ -0.310 - 0.950i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.045020184$$ $$L(\frac12)$$ $$\approx$$ $$2.045020184$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - iT$$
5 $$1 - iT$$
23 $$1 + (4.69 + 0.988i)T$$
good7 $$1 - 1.70T + 7T^{2}$$
11 $$1 - 3.79T + 11T^{2}$$
13 $$1 - 1.61T + 13T^{2}$$
17 $$1 - 2.05iT - 17T^{2}$$
19 $$1 + 0.518T + 19T^{2}$$
29 $$1 + 2.88T + 29T^{2}$$
31 $$1 - 4.16iT - 31T^{2}$$
37 $$1 - 7.56iT - 37T^{2}$$
41 $$1 - 11.8T + 41T^{2}$$
43 $$1 + 4.43T + 43T^{2}$$
47 $$1 - 0.599iT - 47T^{2}$$
53 $$1 + 2.97iT - 53T^{2}$$
59 $$1 - 9.67iT - 59T^{2}$$
61 $$1 + 2.26iT - 61T^{2}$$
67 $$1 - 9.80T + 67T^{2}$$
71 $$1 - 9.51iT - 71T^{2}$$
73 $$1 + 7.31T + 73T^{2}$$
79 $$1 - 7.33T + 79T^{2}$$
83 $$1 - 4.63T + 83T^{2}$$
89 $$1 - 16.0iT - 89T^{2}$$
97 $$1 + 14.4iT - 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

−8.391816309532432933125597094769, −7.79426528662253225138580073006, −6.81411488818707607463921740110, −6.23840875556294053002987353769, −5.52581926154309936954592614402, −4.55231290218120486901089577493, −3.97845864457296593250648723984, −3.26689122278431770145987491542, −2.14694920599887478936678827622, −1.20374872156259294409560338940, 0.56840435121473957590523487527, 1.57179600562302942149221833564, 2.26409217237648921067088437034, 3.57138402560501626586761117882, 4.20585416783454357756129667770, 5.06523025023818978027453056757, 5.93718603647348774969931176917, 6.40006247756074555422595987488, 7.40947284150059640010862886821, 7.83793109686308366454087525600