# Properties

 Label 2-420-60.59-c1-0-40 Degree $2$ Conductor $420$ Sign $0.993 + 0.111i$ Analytic cond. $3.35371$ Root an. cond. $1.83131$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.835 − 1.14i)2-s + (1.10 + 1.33i)3-s + (−0.602 − 1.90i)4-s + (1 + 2i)5-s + (2.44 − 0.140i)6-s + 7-s + (−2.67 − 0.907i)8-s + (−0.569 + 2.94i)9-s + (3.11 + 0.531i)10-s + 2.20·11-s + (1.88 − 2.90i)12-s + 1.89i·13-s + (0.835 − 1.14i)14-s + (−1.56 + 3.54i)15-s + (−3.27 + 2.29i)16-s + 1.13·17-s + ⋯
 L(s)  = 1 + (0.591 − 0.806i)2-s + (0.636 + 0.771i)3-s + (−0.301 − 0.953i)4-s + (0.447 + 0.894i)5-s + (0.998 − 0.0574i)6-s + 0.377·7-s + (−0.947 − 0.320i)8-s + (−0.189 + 0.981i)9-s + (0.985 + 0.167i)10-s + 0.664·11-s + (0.543 − 0.839i)12-s + 0.524i·13-s + (0.223 − 0.304i)14-s + (−0.405 + 0.914i)15-s + (−0.818 + 0.574i)16-s + 0.276·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$420$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.993 + 0.111i$ Analytic conductor: $$3.35371$$ Root analytic conductor: $$1.83131$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{420} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 420,\ (\ :1/2),\ 0.993 + 0.111i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.37537 - 0.132337i$$ $$L(\frac12)$$ $$\approx$$ $$2.37537 - 0.132337i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.835 + 1.14i)T$$
3 $$1 + (-1.10 - 1.33i)T$$
5 $$1 + (-1 - 2i)T$$
7 $$1 - T$$
good11 $$1 - 2.20T + 11T^{2}$$
13 $$1 - 1.89iT - 13T^{2}$$
17 $$1 - 1.13T + 17T^{2}$$
19 $$1 + 8.56iT - 19T^{2}$$
23 $$1 - 3.21iT - 23T^{2}$$
29 $$1 + 1.89iT - 29T^{2}$$
31 $$1 + 5.90iT - 31T^{2}$$
37 $$1 + 0.409iT - 37T^{2}$$
41 $$1 + 4.40iT - 41T^{2}$$
43 $$1 + 0.934T + 43T^{2}$$
47 $$1 + 2.67iT - 47T^{2}$$
53 $$1 + 6.81T + 53T^{2}$$
59 $$1 + 13.5T + 59T^{2}$$
61 $$1 - 12.4T + 61T^{2}$$
67 $$1 + 10.8T + 67T^{2}$$
71 $$1 + 8T + 71T^{2}$$
73 $$1 + 8iT - 73T^{2}$$
79 $$1 - 3.76iT - 79T^{2}$$
83 $$1 - 6.84iT - 83T^{2}$$
89 $$1 + 16.1iT - 89T^{2}$$
97 $$1 - 18.4iT - 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

−11.20822423743695479333261995172, −10.38310199823424405902618039675, −9.469456981305286299950115294100, −8.981225597799270489321245219642, −7.41910979193041177654448983771, −6.25418128538615455074961187735, −5.07436612849118233598200416153, −4.08744286755495355888268657025, −3.03597880506167268154064148991, −2.01413902754149378612222467916, 1.54898627994113520815872536121, 3.23749000233078838930528540860, 4.43315845006297842323589090967, 5.66325777951156073206922414393, 6.40091357015312444951906726912, 7.61103738340364389575417072897, 8.309390418075724492866591321169, 8.955216836775243830040107474914, 10.04110852281352022947695730873, 11.73661535659481206275624809490