# Properties

 Label 2-3360-105.83-c0-0-1 Degree $2$ Conductor $3360$ Sign $-0.850 - 0.525i$ Analytic cond. $1.67685$ Root an. cond. $1.29493$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## 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 + 2i·11-s − 1.00·15-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 + (−1.41 − 1.41i)33-s − 1.00i·35-s + (−1 + i)37-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 + 2i·11-s − 1.00·15-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 + (−1.41 − 1.41i)33-s − 1.00i·35-s + (−1 + i)37-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$3360$$    =    $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.850 - 0.525i$ Analytic conductor: $$1.67685$$ Root analytic conductor: $$1.29493$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3360} (3233, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3360,\ (\ :0),\ -0.850 - 0.525i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7043776608$$ $$L(\frac12)$$ $$\approx$$ $$0.7043776608$$ $$L(1)$$ 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 - 2iT - T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 - 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 - T^{2}$$
73 $$1 - iT^{2}$$
79 $$1 - T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 - 1.41iT - T^{2}$$
97 $$1 + iT^{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

−9.369780317474119681144765835169, −8.563295353373662238432008469777, −7.16275923523562737296268947862, −6.75953727230032611673439721124, −6.37698000436167948237482238110, −5.12502765818217918523432446890, −4.63581582410514797538010248763, −3.72857078874404254294913784706, −2.78534157540500777561613026980, −1.62325491945416098093834723918, 0.44389406066761480172771924739, 1.72102022850308997771877137893, 2.68033255828806444843775289344, 3.72948967295506741411680163173, 5.03781726724227775418070512940, 5.65541706252530411636655841555, 6.11647133080889287714858957776, 6.70818270074833301138431958767, 7.81602273521110090539129012609, 8.729686590010766303149266081473