Properties

 Label 2-2e2-4.3-c6-0-1 Degree $2$ Conductor $4$ Sign $0.875 + 0.484i$ Analytic cond. $0.920216$ Root an. cond. $0.959279$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (2 − 7.74i)2-s + 30.9i·3-s + (−56.0 − 30.9i)4-s + 10·5-s + (240. + 61.9i)6-s − 309. i·7-s + (−352. + 371. i)8-s − 231.·9-s + (20 − 77.4i)10-s + 960. i·11-s + (960. − 1.73e3i)12-s + 1.46e3·13-s + (−2.40e3 − 619. i)14-s + 309. i·15-s + (2.17e3 + 3.47e3i)16-s − 4.76e3·17-s + ⋯
 L(s)  = 1 + (0.250 − 0.968i)2-s + 1.14i·3-s + (−0.875 − 0.484i)4-s + 0.0800·5-s + (1.11 + 0.286i)6-s − 0.903i·7-s + (−0.687 + 0.726i)8-s − 0.316·9-s + (0.0200 − 0.0774i)10-s + 0.721i·11-s + (0.555 − 1.00i)12-s + 0.667·13-s + (−0.874 − 0.225i)14-s + 0.0918i·15-s + (0.531 + 0.847i)16-s − 0.970·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$4$$    =    $$2^{2}$$ Sign: $0.875 + 0.484i$ Analytic conductor: $$0.920216$$ Root analytic conductor: $$0.959279$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{4} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4,\ (\ :3),\ 0.875 + 0.484i)$$

Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$1.00282 - 0.258927i$$ $$L(\frac12)$$ $$\approx$$ $$1.00282 - 0.258927i$$ $$L(4)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-2 + 7.74i)T$$
good3 $$1 - 30.9iT - 729T^{2}$$
5 $$1 - 10T + 1.56e4T^{2}$$
7 $$1 + 309. iT - 1.17e5T^{2}$$
11 $$1 - 960. iT - 1.77e6T^{2}$$
13 $$1 - 1.46e3T + 4.82e6T^{2}$$
17 $$1 + 4.76e3T + 2.41e7T^{2}$$
19 $$1 + 7.52e3iT - 4.70e7T^{2}$$
23 $$1 - 1.04e4iT - 1.48e8T^{2}$$
29 $$1 - 2.54e4T + 5.94e8T^{2}$$
31 $$1 + 4.18e4iT - 8.87e8T^{2}$$
37 $$1 - 1.99e3T + 2.56e9T^{2}$$
41 $$1 - 2.93e4T + 4.75e9T^{2}$$
43 $$1 - 2.15e4iT - 6.32e9T^{2}$$
47 $$1 - 7.56e3iT - 1.07e10T^{2}$$
53 $$1 + 1.92e5T + 2.21e10T^{2}$$
59 $$1 - 7.84e4iT - 4.21e10T^{2}$$
61 $$1 + 1.09e4T + 5.15e10T^{2}$$
67 $$1 - 3.94e5iT - 9.04e10T^{2}$$
71 $$1 + 5.32e5iT - 1.28e11T^{2}$$
73 $$1 - 2.88e5T + 1.51e11T^{2}$$
79 $$1 - 3.10e5iT - 2.43e11T^{2}$$
83 $$1 - 2.04e5iT - 3.26e11T^{2}$$
89 $$1 - 3.10e5T + 4.96e11T^{2}$$
97 $$1 + 1.45e6T + 8.32e11T^{2}$$
show less
$$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

−23.57175299273089970011604651517, −22.19059552843461559609099182859, −20.95756433816106989269764042540, −19.83058320096091245143230269053, −17.59929793768923017837605476704, −15.43249017370527487601311095726, −13.47879660771318523930785751614, −11.03139954962341675621445453554, −9.604181779505845522978754643876, −4.29313366391969956478778604249, 6.27799477609798837982289869787, 8.405706884823255524905660661091, 12.41897331860520707654073360983, 13.89484124930942548130012896258, 15.87856099392868848643883166863, 17.88102778607635228745883699780, 18.90580774121245845310976603894, 21.61858622259975424842452197345, 23.26626467501858284215669449273, 24.53660273464275639743622462145