Properties

Label 2-600-120.53-c1-0-13
Degree $2$
Conductor $600$
Sign $-0.608 - 0.793i$
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  + (−1.18 + 0.771i)2-s + (−0.713 + 1.57i)3-s + (0.808 − 1.82i)4-s + (−0.373 − 2.42i)6-s + (1.44 − 1.44i)7-s + (0.454 + 2.79i)8-s + (−1.98 − 2.25i)9-s − 0.641·11-s + (2.31 + 2.58i)12-s + (−2.03 + 2.03i)13-s + (−0.597 + 2.83i)14-s + (−2.69 − 2.95i)16-s + (4.37 + 4.37i)17-s + (4.08 + 1.13i)18-s + 4.93·19-s + ⋯
L(s)  = 1  + (−0.837 + 0.545i)2-s + (−0.411 + 0.911i)3-s + (0.404 − 0.914i)4-s + (−0.152 − 0.988i)6-s + (0.546 − 0.546i)7-s + (0.160 + 0.987i)8-s + (−0.660 − 0.750i)9-s − 0.193·11-s + (0.667 + 0.744i)12-s + (−0.563 + 0.563i)13-s + (−0.159 + 0.756i)14-s + (−0.673 − 0.739i)16-s + (1.06 + 1.06i)17-s + (0.963 + 0.267i)18-s + 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337933 + 0.684451i\)
\(L(\frac12)\) \(\approx\) \(0.337933 + 0.684451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.771i)T \)
3 \( 1 + (0.713 - 1.57i)T \)
5 \( 1 \)
good7 \( 1 + (-1.44 + 1.44i)T - 7iT^{2} \)
11 \( 1 + 0.641T + 11T^{2} \)
13 \( 1 + (2.03 - 2.03i)T - 13iT^{2} \)
17 \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 + (3.73 - 3.73i)T - 23iT^{2} \)
29 \( 1 - 9.84iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + (3.44 + 3.44i)T + 37iT^{2} \)
41 \( 1 - 7.20iT - 41T^{2} \)
43 \( 1 + (4.37 - 4.37i)T - 43iT^{2} \)
47 \( 1 + (4.08 + 4.08i)T + 47iT^{2} \)
53 \( 1 + (3.83 + 3.83i)T + 53iT^{2} \)
59 \( 1 + 2.50iT - 59T^{2} \)
61 \( 1 - 9.64iT - 61T^{2} \)
67 \( 1 + (2.59 + 2.59i)T + 67iT^{2} \)
71 \( 1 + 16.7iT - 71T^{2} \)
73 \( 1 + (-8.40 - 8.40i)T + 73iT^{2} \)
79 \( 1 - 5.31iT - 79T^{2} \)
83 \( 1 + (-2.20 - 2.20i)T + 83iT^{2} \)
89 \( 1 - 3.96T + 89T^{2} \)
97 \( 1 + (1.11 - 1.11i)T - 97iT^{2} \)
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

−10.70072850825035736168939085276, −10.00596887019663250892076947921, −9.422801115012557570620162526010, −8.323780813193599043346799331941, −7.58060699462671057921815619458, −6.51612859325352973405013547884, −5.47278133429323342891990497797, −4.75674298176205414946151890520, −3.37972888396975367689346154958, −1.40084394594113568039876482172, 0.63946366062939612657197817199, 2.10458970998954451881143411042, 3.05335237750847854934003281696, 4.88918163249188192662647496142, 5.89727044329944014242133392168, 7.09466888747597281180262950259, 7.86345112883045301953803383940, 8.367486908369380084386009924563, 9.647983519038996219857624042263, 10.29039538964228957875226619901

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