Properties

 Degree $2$ Conductor $600$ Sign $-0.447 - 0.894i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + i·3-s − 9-s − 4·11-s + 6i·13-s + 6i·17-s + 4·19-s − i·27-s + 2·29-s − 8·31-s − 4i·33-s + 2i·37-s − 6·39-s − 6·41-s + 12i·43-s − 8i·47-s + ⋯
 L(s)  = 1 + 0.577i·3-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s + 1.45i·17-s + 0.917·19-s − 0.192i·27-s + 0.371·29-s − 1.43·31-s − 0.696i·33-s + 0.328i·37-s − 0.960·39-s − 0.937·41-s + 1.82i·43-s − 1.16i·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$600$$    =    $$2^{3} \cdot 3 \cdot 5^{2}$$ Sign: $-0.447 - 0.894i$ Motivic weight: $$1$$ Character: $\chi_{600} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 600,\ (\ :1/2),\ -0.447 - 0.894i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.567304 + 0.917917i$$ $$L(\frac12)$$ $$\approx$$ $$0.567304 + 0.917917i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - iT$$
5 $$1$$
good7 $$1 - 7T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 - 6iT - 13T^{2}$$
17 $$1 - 6iT - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 + 8T + 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 - 12iT - 43T^{2}$$
47 $$1 + 8iT - 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 - 14T + 61T^{2}$$
67 $$1 + 4iT - 67T^{2}$$
71 $$1 - 8T + 71T^{2}$$
73 $$1 + 6iT - 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + 12iT - 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 2iT - 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

−10.86818693027999930272164856398, −10.10520309252518321572473208349, −9.263308763439584118626108863212, −8.417456432206742402616774187403, −7.46441289009484144811939004072, −6.37253867183982780033524770776, −5.35248285851307466949531522148, −4.40236227739555376033432314237, −3.35159111890061679349962992986, −1.91768409377280683847734840880, 0.58653194341292848366378504939, 2.45504709485535561430260517619, 3.36608825253678448957738772155, 5.19886710852154865813871202859, 5.53888550831681015812030719963, 7.05659105140906665667498554743, 7.63359375554363916623389748321, 8.447633664682078280223357507142, 9.569372257466946800048741667272, 10.43717036399489941850520662707