# Properties

 Label 2-3360-5.4-c1-0-69 Degree $2$ Conductor $3360$ Sign $-0.894 - 0.447i$ Analytic cond. $26.8297$ Root an. cond. $5.17974$ 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 + (1 − 2i)5-s − i·7-s − 9-s + 2·11-s + 2i·13-s + (−2 − i)15-s − 6·19-s − 21-s + 8i·23-s + (−3 − 4i)25-s + i·27-s − 6·29-s − 10·31-s − 2i·33-s + ⋯
 L(s)  = 1 − 0.577i·3-s + (0.447 − 0.894i)5-s − 0.377i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (−0.516 − 0.258i)15-s − 1.37·19-s − 0.218·21-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.192i·27-s − 1.11·29-s − 1.79·31-s − 0.348i·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3360$$    =    $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.894 - 0.447i$ Analytic conductor: $$26.8297$$ Root analytic conductor: $$5.17974$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3360} (2689, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3360,\ (\ :1/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5551001197$$ $$L(\frac12)$$ $$\approx$$ $$0.5551001197$$ $$L(\frac{3}{2})$$ 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 + (-1 + 2i)T$$
7 $$1 + iT$$
good11 $$1 - 2T + 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 + 6T + 19T^{2}$$
23 $$1 - 8iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + 10T + 31T^{2}$$
37 $$1 + 8iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 + 8iT - 47T^{2}$$
53 $$1 + 2iT - 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 - 14T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 2T + 71T^{2}$$
73 $$1 - 2iT - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 83T^{2}$$
89 $$1 + 6T + 89T^{2}$$
97 $$1 + 14iT - 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

−8.274512137559448455460535201643, −7.31629850307708500964429993834, −6.85000642018443407104843059337, −5.79431844707338327410518512145, −5.37476953136403275462481248555, −4.18834287360190066692406421460, −3.63499789754054518610993517307, −2.01100176092952639044898944376, −1.59462033990013488894225854547, −0.15157138060999760872668201614, 1.78303011523515690794133173816, 2.69092832449966498426826711707, 3.51718245214933408380273794475, 4.37898966101054919940999309713, 5.29514266980052317167154322762, 6.17629148876581429129953919448, 6.57160460943585065057136817172, 7.55628711068129622695664873114, 8.448031672138903000288655699172, 9.077982012850391101525928114207