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Properties

Label 2-600-120.53-c1-0-4
Degree $2$
Conductor $600$
Sign $-0.999 + 0.0248i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.313 + 1.37i)2-s + (−1.69 − 0.378i)3-s + (−1.80 − 0.865i)4-s + (1.05 − 2.21i)6-s + (−0.699 + 0.699i)7-s + (1.75 − 2.21i)8-s + (2.71 + 1.27i)9-s + 4.24·11-s + (2.71 + 2.14i)12-s + (−2.08 + 2.08i)13-s + (−0.745 − 1.18i)14-s + (2.50 + 3.12i)16-s + (0.0541 + 0.0541i)17-s + (−2.61 + 3.33i)18-s − 4.52·19-s + ⋯
L(s)  = 1  + (−0.221 + 0.975i)2-s + (−0.975 − 0.218i)3-s + (−0.901 − 0.432i)4-s + (0.429 − 0.902i)6-s + (−0.264 + 0.264i)7-s + (0.621 − 0.783i)8-s + (0.904 + 0.426i)9-s + 1.27·11-s + (0.785 + 0.619i)12-s + (−0.577 + 0.577i)13-s + (−0.199 − 0.316i)14-s + (0.625 + 0.780i)16-s + (0.0131 + 0.0131i)17-s + (−0.616 + 0.787i)18-s − 1.03·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.999 + 0.0248i$
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: \((2,\ 600,\ (\ :1/2),\ -0.999 + 0.0248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00512955 - 0.412097i\)
\(L(\frac12)\) \(\approx\) \(0.00512955 - 0.412097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.313 - 1.37i)T \)
3 \( 1 + (1.69 + 0.378i)T \)
5 \( 1 \)
good7 \( 1 + (0.699 - 0.699i)T - 7iT^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (2.08 - 2.08i)T - 13iT^{2} \)
17 \( 1 + (-0.0541 - 0.0541i)T + 17iT^{2} \)
19 \( 1 + 4.52T + 19T^{2} \)
23 \( 1 + (3.48 - 3.48i)T - 23iT^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (-7.29 - 7.29i)T + 37iT^{2} \)
41 \( 1 - 8.74iT - 41T^{2} \)
43 \( 1 + (4.60 - 4.60i)T - 43iT^{2} \)
47 \( 1 + (8.05 + 8.05i)T + 47iT^{2} \)
53 \( 1 + (0.260 + 0.260i)T + 53iT^{2} \)
59 \( 1 + 2.89iT - 59T^{2} \)
61 \( 1 - 6.70iT - 61T^{2} \)
67 \( 1 + (1.75 + 1.75i)T + 67iT^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (4.53 + 4.53i)T + 73iT^{2} \)
79 \( 1 - 1.46iT - 79T^{2} \)
83 \( 1 + (-5.61 - 5.61i)T + 83iT^{2} \)
89 \( 1 + 5.42T + 89T^{2} \)
97 \( 1 + (-11.3 + 11.3i)T - 97iT^{2} \)
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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

−11.19295545023151899713060307520, −9.901720127299004182074908141020, −9.482948886738631297187394779656, −8.338682944110379717052979295765, −7.32553075173095577744506763800, −6.47926267538483736255072289474, −5.99682491767211414307431982665, −4.81061486516977679711030852227, −3.97305492287448728963883224500, −1.59153137168970981184291833551, 0.29101439994378098528462496356, 1.87244970800108041084400656997, 3.60689698453807034223202211679, 4.34668687970278153053671874208, 5.45491599568766111772300611346, 6.55187100480986576312088919192, 7.56052899501062349838508434990, 8.861613587791413745691355131884, 9.561085349844781160171201917899, 10.45475305582353703174643255409

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