↑

Properties

Label 2-600-15.8-c1-0-2
Degree $2$
Conductor $600$
Sign $-0.296 - 0.954i$
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  + (−1.73 − 0.0537i)3-s + (−0.560 + 0.560i)7-s + (2.99 + 0.186i)9-s + 0.939i·11-s + (−4.13 − 4.13i)13-s + (4.50 + 4.50i)17-s + 3.74i·19-s + (0.999 − 0.939i)21-s + (−6.18 + 6.18i)23-s + (−5.17 − 0.483i)27-s + 3.16·29-s − 5.37·31-s + (0.0505 − 1.62i)33-s + (−3.56 + 3.56i)37-s + (6.92 + 7.37i)39-s + ⋯
L(s)  = 1  + (−0.999 − 0.0310i)3-s + (−0.211 + 0.211i)7-s + (0.998 + 0.0620i)9-s + 0.283i·11-s + (−1.14 − 1.14i)13-s + (1.09 + 1.09i)17-s + 0.859i·19-s + (0.218 − 0.205i)21-s + (−1.28 + 1.28i)23-s + (−0.995 − 0.0929i)27-s + 0.588·29-s − 0.964·31-s + (0.00879 − 0.283i)33-s + (−0.586 + 0.586i)37-s + (1.10 + 1.18i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369847 + 0.502312i\)
\(L(\frac12)\) \(\approx\) \(0.369847 + 0.502312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0537i)T \)
5 \( 1 \)
good7 \( 1 + (0.560 - 0.560i)T - 7iT^{2} \)
11 \( 1 - 0.939iT - 11T^{2} \)
13 \( 1 + (4.13 + 4.13i)T + 13iT^{2} \)
17 \( 1 + (-4.50 - 4.50i)T + 17iT^{2} \)
19 \( 1 - 3.74iT - 19T^{2} \)
23 \( 1 + (6.18 - 6.18i)T - 23iT^{2} \)
29 \( 1 - 3.16T + 29T^{2} \)
31 \( 1 + 5.37T + 31T^{2} \)
37 \( 1 + (3.56 - 3.56i)T - 37iT^{2} \)
41 \( 1 - 9.15iT - 41T^{2} \)
43 \( 1 + (-4.33 - 4.33i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (-0.526 + 0.526i)T - 53iT^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 1.37T + 61T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - 67iT^{2} \)
71 \( 1 + 3.16iT - 71T^{2} \)
73 \( 1 + (5.56 + 5.56i)T + 73iT^{2} \)
79 \( 1 + 8.74iT - 79T^{2} \)
83 \( 1 + (-3.97 + 3.97i)T - 83iT^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + (-5.25 + 5.25i)T - 97iT^{2} \)
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

−10.79956888975129715578866566961, −10.09983045451581221138731234834, −9.570209032592891462413540729517, −7.928493806250541560555941765790, −7.52585106789840871737199098313, −6.08896650708405909956335093347, −5.65213421832659108046169385777, −4.53926529916402452331745987155, −3.28502536162466829115230598709, −1.55347246179317140807096851578, 0.39816284654743089177016667293, 2.27581769526743710398687924032, 3.93936013720005987570166071151, 4.89913891459098051499699030543, 5.76349068816329627173559398720, 6.94460937865581089241977965569, 7.32781436082071482797330317856, 8.814492479808315993359505619613, 9.738608667561174430450145559662, 10.39107812085426003903863653954

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