# Properties

 Label 2-120-120.59-c1-0-4 Degree $2$ Conductor $120$ Sign $0.395 - 0.918i$ Analytic cond. $0.958204$ Root an. cond. $0.978879$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 1.11i)2-s + 1.73·3-s + (−0.500 − 1.93i)4-s + 2.23i·5-s + (−1.49 + 1.93i)6-s + (2.59 + 1.11i)8-s + 2.99·9-s + (−2.50 − 1.93i)10-s + (−0.866 − 3.35i)12-s + 3.87i·15-s + (−3.5 + 1.93i)16-s − 6.92·17-s + (−2.59 + 3.35i)18-s + 4·19-s + (4.33 − 1.11i)20-s + ⋯
 L(s)  = 1 + (−0.612 + 0.790i)2-s + 1.00·3-s + (−0.250 − 0.968i)4-s + 0.999i·5-s + (−0.612 + 0.790i)6-s + (0.918 + 0.395i)8-s + 0.999·9-s + (−0.790 − 0.612i)10-s + (−0.250 − 0.968i)12-s + 1.00i·15-s + (−0.875 + 0.484i)16-s − 1.68·17-s + (−0.612 + 0.790i)18-s + 0.917·19-s + (0.968 − 0.250i)20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$120$$    =    $$2^{3} \cdot 3 \cdot 5$$ Sign: $0.395 - 0.918i$ Analytic conductor: $$0.958204$$ Root analytic conductor: $$0.978879$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{120} (59, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 120,\ (\ :1/2),\ 0.395 - 0.918i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.867349 + 0.571002i$$ $$L(\frac12)$$ $$\approx$$ $$0.867349 + 0.571002i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 1.11i)T$$
3 $$1 - 1.73T$$
5 $$1 - 2.23iT$$
good7 $$1 + 7T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + 6.92T + 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + 8.94iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 7.74iT - 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 - 8.94iT - 47T^{2}$$
53 $$1 + 4.47iT - 53T^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 - 15.4iT - 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 7.74iT - 79T^{2}$$
83 $$1 + 3.46T + 83T^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 - 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

−14.01057140315348713047290722019, −13.10697944041546996355999029979, −11.23860334373571340308666406775, −10.24684930752074628167329776622, −9.287519628760410954942504180246, −8.252627641704206271703064071715, −7.21442206638901471156076448920, −6.33051875775614412315787853556, −4.37537421501178733384641744417, −2.44853504103337796203918483388, 1.72819469709323362843672116265, 3.46359795232067585996238519145, 4.79833409731623754548422130776, 7.18814680219832983582668286538, 8.293006408197617641650026815652, 9.100050428973946658486016822351, 9.808524879053431654339438705285, 11.21527244186148830221963879333, 12.30765025102694088948430795586, 13.27963007760345288276800128713