# Properties

 Label 2-74-37.36-c1-0-2 Degree $2$ Conductor $74$ Sign $0.753 + 0.657i$ Analytic cond. $0.590892$ Root an. cond. $0.768695$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + 1.79·3-s − 4-s + 0.791i·5-s − 1.79i·6-s − 2·7-s + i·8-s + 0.208·9-s + 0.791·10-s − 0.791·11-s − 1.79·12-s − 3.79i·13-s + 2i·14-s + 1.41i·15-s + 16-s + 7.58i·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 1.03·3-s − 0.5·4-s + 0.353i·5-s − 0.731i·6-s − 0.755·7-s + 0.353i·8-s + 0.0695·9-s + 0.250·10-s − 0.238·11-s − 0.517·12-s − 1.05i·13-s + 0.534i·14-s + 0.365i·15-s + 0.250·16-s + 1.83i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$74$$    =    $$2 \cdot 37$$ Sign: $0.753 + 0.657i$ Analytic conductor: $$0.590892$$ Root analytic conductor: $$0.768695$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{74} (73, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 74,\ (\ :1/2),\ 0.753 + 0.657i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.01092 - 0.379142i$$ $$L(\frac12)$$ $$\approx$$ $$1.01092 - 0.379142i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
37 $$1 + (-4 + 4.58i)T$$
good3 $$1 - 1.79T + 3T^{2}$$
5 $$1 - 0.791iT - 5T^{2}$$
7 $$1 + 2T + 7T^{2}$$
11 $$1 + 0.791T + 11T^{2}$$
13 $$1 + 3.79iT - 13T^{2}$$
17 $$1 - 7.58iT - 17T^{2}$$
19 $$1 - 1.58iT - 19T^{2}$$
23 $$1 + 3.79iT - 23T^{2}$$
29 $$1 - 3.79iT - 29T^{2}$$
31 $$1 + 8.37iT - 31T^{2}$$
41 $$1 - 9.79T + 41T^{2}$$
43 $$1 + 6iT - 43T^{2}$$
47 $$1 + 7.58T + 47T^{2}$$
53 $$1 + 1.58T + 53T^{2}$$
59 $$1 - 1.58iT - 59T^{2}$$
61 $$1 - 12.7iT - 61T^{2}$$
67 $$1 - 6.37T + 67T^{2}$$
71 $$1 + 9.16T + 71T^{2}$$
73 $$1 + 4.37T + 73T^{2}$$
79 $$1 - 8.20iT - 79T^{2}$$
83 $$1 - 15.1T + 83T^{2}$$
89 $$1 - 6iT - 89T^{2}$$
97 $$1 + 13.5iT - 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

−14.51857667129917132603272307751, −13.14215444081536577234990046141, −12.64020750102322277673949082459, −10.91633864637183167844453857725, −10.04927674140541890680663199485, −8.822974507471760822583719598981, −7.84117264190993255120497014889, −5.97832402100446085088432750213, −3.77761671350164625409554474866, −2.62509516017638999425379875924, 3.04893253332420279706449546079, 4.84351231959469134717944299118, 6.59542030368139108942676692192, 7.78092525267977142108962169786, 9.079728672022026253730412723331, 9.566375257743002105551904355025, 11.53096802022679387544575129822, 12.98493668412561973418339308322, 13.83156345533807426231772609100, 14.57429349606826310858075089623