↑

Properties

Label 2-600-120.53-c1-0-11
Degree $2$
Conductor $600$
Sign $0.968 - 0.247i$
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  + (−1.11 − 0.864i)2-s + (−0.170 − 1.72i)3-s + (0.506 + 1.93i)4-s + (−1.29 + 2.07i)6-s + (−2.06 + 2.06i)7-s + (1.10 − 2.60i)8-s + (−2.94 + 0.586i)9-s − 0.510·11-s + (3.24 − 1.20i)12-s + (−0.750 + 0.750i)13-s + (4.10 − 0.528i)14-s + (−3.48 + 1.95i)16-s + (3.14 + 3.14i)17-s + (3.80 + 1.88i)18-s + 6.01·19-s + ⋯
L(s)  = 1  + (−0.791 − 0.611i)2-s + (−0.0982 − 0.995i)3-s + (0.253 + 0.967i)4-s + (−0.530 + 0.847i)6-s + (−0.782 + 0.782i)7-s + (0.390 − 0.920i)8-s + (−0.980 + 0.195i)9-s − 0.153·11-s + (0.937 − 0.346i)12-s + (−0.208 + 0.208i)13-s + (1.09 − 0.141i)14-s + (−0.871 + 0.489i)16-s + (0.763 + 0.763i)17-s + (0.895 + 0.444i)18-s + 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.968 - 0.247i$
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: \((2,\ 600,\ (\ :1/2),\ 0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657381 + 0.0827049i\)
\(L(\frac12)\) \(\approx\) \(0.657381 + 0.0827049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.864i)T \)
3 \( 1 + (0.170 + 1.72i)T \)
5 \( 1 \)
good7 \( 1 + (2.06 - 2.06i)T - 7iT^{2} \)
11 \( 1 + 0.510T + 11T^{2} \)
13 \( 1 + (0.750 - 0.750i)T - 13iT^{2} \)
17 \( 1 + (-3.14 - 3.14i)T + 17iT^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + (2.54 - 2.54i)T - 23iT^{2} \)
29 \( 1 - 5.10iT - 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (-6.76 - 6.76i)T + 37iT^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + (-5.95 + 5.95i)T - 43iT^{2} \)
47 \( 1 + (3.33 + 3.33i)T + 47iT^{2} \)
53 \( 1 + (-5.75 - 5.75i)T + 53iT^{2} \)
59 \( 1 - 1.16iT - 59T^{2} \)
61 \( 1 + 4.92iT - 61T^{2} \)
67 \( 1 + (7.98 + 7.98i)T + 67iT^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 + (-3.20 - 3.20i)T + 73iT^{2} \)
79 \( 1 - 7.31iT - 79T^{2} \)
83 \( 1 + (4.77 + 4.77i)T + 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (10.8 - 10.8i)T - 97iT^{2} \)
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

−10.72097197929394046490179091183, −9.698375513009959923618120092400, −9.049715414199463392028129664126, −8.046274447824078978367083218852, −7.37787516418525158429056273727, −6.37557535509800367319042361840, −5.43004044892662659950632049519, −3.50973953098499879040113903958, −2.61115856606044233159077094192, −1.32531988825101389783185633991, 0.52500608641339845628884371340, 2.82917406314031901680434882736, 4.10480643815372305375659541160, 5.30200314773194355250236778908, 6.05112900344338077404023446722, 7.25011219928665630035624807131, 7.916323761624280671218558022330, 9.186486808029409176043408337902, 9.717564630943050612269133012001, 10.28349086774241583464413923606

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