Properties

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

Origins

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Normalization:  

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

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

Graph of the $Z$-function along the critical line