# Properties

 Label 2-3360-105.62-c0-0-1 Degree $2$ Conductor $3360$ Sign $-0.525 - 0.850i$ Analytic cond. $1.67685$ Root an. cond. $1.29493$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s + 1.00i·15-s + (−1.41 − 1.41i)17-s − 1.41·19-s − 1.00·21-s + (1 + i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·31-s − 1.00·35-s + (1 + i)37-s + 1.41·41-s + ⋯
 L(s)  = 1 + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s + 1.00i·15-s + (−1.41 − 1.41i)17-s − 1.41·19-s − 1.00·21-s + (1 + i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·31-s − 1.00·35-s + (1 + i)37-s + 1.41·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3360$$    =    $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.525 - 0.850i$ Analytic conductor: $$1.67685$$ Root analytic conductor: $$1.29493$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3360} (1217, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3360,\ (\ :0),\ -0.525 - 0.850i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.475945180$$ $$L(\frac12)$$ $$\approx$$ $$1.475945180$$ $$L(1)$$ 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 + (-0.707 - 0.707i)T$$
5 $$1 + (-0.707 - 0.707i)T$$
7 $$1 + (0.707 - 0.707i)T$$
good11 $$1 - T^{2}$$
13 $$1 + iT^{2}$$
17 $$1 + (1.41 + 1.41i)T + iT^{2}$$
19 $$1 + 1.41T + T^{2}$$
23 $$1 + (-1 - i)T + iT^{2}$$
29 $$1 + T^{2}$$
31 $$1 - 1.41iT - T^{2}$$
37 $$1 + (-1 - i)T + iT^{2}$$
41 $$1 - 1.41T + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + iT^{2}$$
71 $$1 + 2iT - T^{2}$$
73 $$1 + iT^{2}$$
79 $$1 - T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + 1.41iT - T^{2}$$
97 $$1 - iT^{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

−9.152762337271154083849193637719, −8.647188616933480927540422401622, −7.51261322385210389200016227239, −6.77285201365310256102065864059, −6.12922659319171642054425028512, −5.15444193027023143104803891148, −4.47406885138366541483869650523, −3.26446227484495128333845277901, −2.74538573981288335019967272345, −1.99740591748223141957097562317, 0.75864946296297369517950204106, 2.03515831636750882701375497583, 2.62247524080756529876846945993, 4.10377789202258324898164171755, 4.29412392941798015255737200479, 5.87555554137069728187084209125, 6.35295122920641661242240613270, 6.95052263867316841114402604751, 7.939208453326992675755090436258, 8.608964089648895358324878786181