# Properties

 Label 2-5520-92.91-c1-0-31 Degree $2$ Conductor $5520$ Sign $-0.157 - 0.987i$ Analytic cond. $44.0774$ Root an. cond. $6.63908$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + i·5-s − 4.68·7-s − 9-s + 3.83·11-s − 2.63·13-s − 15-s − 5.43i·17-s + 4.36·19-s − 4.68i·21-s + (−4.47 − 1.71i)23-s − 25-s − i·27-s + 6.33·29-s + 4.22i·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.447i·5-s − 1.76·7-s − 0.333·9-s + 1.15·11-s − 0.731·13-s − 0.258·15-s − 1.31i·17-s + 1.00·19-s − 1.02i·21-s + (−0.933 − 0.357i)23-s − 0.200·25-s − 0.192i·27-s + 1.17·29-s + 0.758i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5520$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 23$$ Sign: $-0.157 - 0.987i$ 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: $$(2,\ 5520,\ (\ :1/2),\ -0.157 - 0.987i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.174185281$$ $$L(\frac12)$$ $$\approx$$ $$1.174185281$$ $$L(\frac{3}{2})$$ 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 + (4.47 + 1.71i)T$$
good7 $$1 + 4.68T + 7T^{2}$$
11 $$1 - 3.83T + 11T^{2}$$
13 $$1 + 2.63T + 13T^{2}$$
17 $$1 + 5.43iT - 17T^{2}$$
19 $$1 - 4.36T + 19T^{2}$$
29 $$1 - 6.33T + 29T^{2}$$
31 $$1 - 4.22iT - 31T^{2}$$
37 $$1 + 5.00iT - 37T^{2}$$
41 $$1 + 5.60T + 41T^{2}$$
43 $$1 - 12.1T + 43T^{2}$$
47 $$1 - 5.14iT - 47T^{2}$$
53 $$1 - 2.61iT - 53T^{2}$$
59 $$1 - 1.55iT - 59T^{2}$$
61 $$1 - 11.4iT - 61T^{2}$$
67 $$1 - 12.6T + 67T^{2}$$
71 $$1 - 12.0iT - 71T^{2}$$
73 $$1 + 13.3T + 73T^{2}$$
79 $$1 - 3.50T + 79T^{2}$$
83 $$1 + 8.01T + 83T^{2}$$
89 $$1 + 15.9iT - 89T^{2}$$
97 $$1 - 10.5iT - 97T^{2}$$
show less
$$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

−8.570966339339210307180243644093, −7.25243867728329851067979116945, −7.06098109919440875464189401992, −6.18710074313132651937199391299, −5.62193669537407312384524945436, −4.55271658276631764501722341362, −3.84503860413908393223579716295, −3.03325218062673289400060852466, −2.56646965201117672858558356724, −0.829124891620464699039810687028, 0.41224490574950001906979117831, 1.50500518523833353849120067361, 2.57027395182130708757847549299, 3.51506153880787033688962620008, 4.03976767764865179536802497945, 5.18665010815259070975271503072, 6.14469789304046635756266322580, 6.38359229846770043129904538468, 7.14655708614863914782249166584, 7.930602738812864076093703265808