Properties

Label 2-600-5.4-c1-0-4
Degree $2$
Conductor $600$
Sign $0.447 - 0.894i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 9-s + 4·11-s + 2i·13-s + 2i·17-s + 4·19-s + 8i·23-s i·27-s − 6·29-s + 8·31-s + 4i·33-s + 6i·37-s − 2·39-s − 6·41-s − 4i·43-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + 0.485i·17-s + 0.917·19-s + 1.66i·23-s − 0.192i·27-s − 1.11·29-s + 1.43·31-s + 0.696i·33-s + 0.986i·37-s − 0.320·39-s − 0.937·41-s − 0.609i·43-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: $0.447 - 0.894i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(1.28259 + 0.792687i\)
\(L(\frac12)\) \(\approx\) \(1.28259 + 0.792687i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
show more
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

−10.84689869069326518718596802165, −9.722755076682735790108006807249, −9.319980256554947494896591516093, −8.305801328331706763494157577162, −7.22689213984221627371390229595, −6.25742097909589294180313204260, −5.25755492369751829160969529345, −4.13023788952651458450579076642, −3.29839074888651189590309568332, −1.55789354773542048252640331200, 0.959303705450604034011833009324, 2.50167583361520930081768631471, 3.75594847300271695208947856097, 5.00001355591556398908592636892, 6.12870744714571890620967147062, 6.89625889003198805119507742819, 7.81492004770371017925162683408, 8.744876854152440905642778675731, 9.555110999509083129577208498245, 10.55040768262254683950084753932

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