Properties

 Label 2-120-120.59-c1-0-14 Degree $2$ Conductor $120$ Sign $0.965 + 0.259i$ 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 + (1.30 + 0.541i)2-s + (−0.541 − 1.64i)3-s + (1.41 + 1.41i)4-s + (−1.25 − 1.84i)5-s + (0.183 − 2.44i)6-s + 3.29·7-s + (1.08 + 2.61i)8-s + (−2.41 + 1.78i)9-s + (−0.645 − 3.09i)10-s + 2.51i·11-s + (1.56 − 3.09i)12-s − 4.65·13-s + (4.29 + 1.78i)14-s + (−2.35 + 3.07i)15-s + 4i·16-s − 3.69·17-s + ⋯
 L(s)  = 1 + (0.923 + 0.382i)2-s + (−0.312 − 0.949i)3-s + (0.707 + 0.707i)4-s + (−0.563 − 0.826i)5-s + (0.0748 − 0.997i)6-s + 1.24·7-s + (0.382 + 0.923i)8-s + (−0.804 + 0.593i)9-s + (−0.204 − 0.978i)10-s + 0.759i·11-s + (0.450 − 0.892i)12-s − 1.29·13-s + (1.14 + 0.475i)14-s + (−0.609 + 0.793i)15-s + i·16-s − 0.896·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$120$$    =    $$2^{3} \cdot 3 \cdot 5$$ Sign: $0.965 + 0.259i$ 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: $$(2,\ 120,\ (\ :1/2),\ 0.965 + 0.259i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.51367 - 0.199512i$$ $$L(\frac12)$$ $$\approx$$ $$1.51367 - 0.199512i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.30 - 0.541i)T$$
3 $$1 + (0.541 + 1.64i)T$$
5 $$1 + (1.25 + 1.84i)T$$
good7 $$1 - 3.29T + 7T^{2}$$
11 $$1 - 2.51iT - 11T^{2}$$
13 $$1 + 4.65T + 13T^{2}$$
17 $$1 + 3.69T + 17T^{2}$$
19 $$1 - 0.828T + 19T^{2}$$
23 $$1 + 2.61iT - 23T^{2}$$
29 $$1 - 6.08T + 29T^{2}$$
31 $$1 - 1.17iT - 31T^{2}$$
37 $$1 + 1.92T + 37T^{2}$$
41 $$1 + 8.59iT - 41T^{2}$$
43 $$1 + 6.01iT - 43T^{2}$$
47 $$1 - 2.61iT - 47T^{2}$$
53 $$1 + 4.59iT - 53T^{2}$$
59 $$1 + 2.51iT - 59T^{2}$$
61 $$1 - 8.48iT - 61T^{2}$$
67 $$1 - 3.29iT - 67T^{2}$$
71 $$1 - 7.12T + 71T^{2}$$
73 $$1 + 6.58iT - 73T^{2}$$
79 $$1 + 16.4iT - 79T^{2}$$
83 $$1 - 9.37T + 83T^{2}$$
89 $$1 - 5.03iT - 89T^{2}$$
97 $$1 - 2.72iT - 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

−13.37558729152493185464555047848, −12.19516989166445017328410954975, −12.05662815796850838539957792052, −10.83524102693976651293622939478, −8.640897529960947201732920776977, −7.72961979167501126242858879964, −6.91638926960422020293116211718, −5.22426025386522193083965894859, −4.55646788193322558451110655672, −2.12174215673087104247097525244, 2.82304005995032521481622582497, 4.26023846643999984491931086043, 5.17618412761611156655762021222, 6.59885521944234893067007730064, 8.042207304822723037877938415520, 9.736694185519729921740280487918, 10.85866047061292968539487998328, 11.35062573991602776134712403822, 12.13715191625321046908267293130, 13.84708498679447777135025673803