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Properties

Label 2-600-5.4-c7-0-48
Degree $2$
Conductor $600$
Sign $0.447 + 0.894i$
Analytic cond. $187.431$
Root an. cond. $13.6905$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27i·3-s − 504i·7-s − 729·9-s + 3.81e3·11-s + 9.57e3i·13-s − 2.60e4i·17-s + 3.83e4·19-s + 1.36e4·21-s − 7.11e4i·23-s − 1.96e4i·27-s − 7.42e4·29-s − 2.75e5·31-s + 1.02e5i·33-s + 2.66e5i·37-s − 2.58e5·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.555i·7-s − 0.333·9-s + 0.863·11-s + 1.20i·13-s − 1.28i·17-s + 1.28·19-s + 0.320·21-s − 1.21i·23-s − 0.192i·27-s − 0.565·29-s − 1.66·31-s + 0.498i·33-s + 0.865i·37-s − 0.697·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(187.431\)
Root analytic conductor: \(13.6905\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :7/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.798715280\)
\(L(\frac12)\) \(\approx\) \(1.798715280\)
\(L(\frac{9}{2})\) 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 \( 1 - 27iT \)
5 \( 1 \)
good7 \( 1 + 504iT - 8.23e5T^{2} \)
11 \( 1 - 3.81e3T + 1.94e7T^{2} \)
13 \( 1 - 9.57e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.60e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.83e4T + 8.93e8T^{2} \)
23 \( 1 + 7.11e4iT - 3.40e9T^{2} \)
29 \( 1 + 7.42e4T + 1.72e10T^{2} \)
31 \( 1 + 2.75e5T + 2.75e10T^{2} \)
37 \( 1 - 2.66e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.84e5T + 1.94e11T^{2} \)
43 \( 1 - 2.45e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.78e5iT - 5.06e11T^{2} \)
53 \( 1 + 5.69e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.52e6T + 2.48e12T^{2} \)
61 \( 1 + 2.64e6T + 3.14e12T^{2} \)
67 \( 1 + 1.41e6iT - 6.06e12T^{2} \)
71 \( 1 + 3.51e6T + 9.09e12T^{2} \)
73 \( 1 - 4.73e6iT - 1.10e13T^{2} \)
79 \( 1 + 4.66e6T + 1.92e13T^{2} \)
83 \( 1 + 5.72e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.19e7T + 4.42e13T^{2} \)
97 \( 1 + 7.15e6iT - 8.07e13T^{2} \)
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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

−9.367300472474562533358670088482, −8.827910671563776059768094543270, −7.43916251976490443872808580599, −6.85463506165866628274147531397, −5.69061744073418056884554134961, −4.61499403499016270572858525093, −3.92989524971167734534060991116, −2.84001941120699009691344812448, −1.50637904505272312793679358633, −0.36805017783066326253133837463, 0.976687529344240556355495197289, 1.83219119473293629269526047490, 3.07876579308734491099830021451, 3.96030383693974213127401430265, 5.65678239707707286266305127878, 5.77219296543278303600907921956, 7.23590740110146610051688680840, 7.77708730511469745471234011518, 8.879475130471737418369162004332, 9.498341693621220550811978807955

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