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Properties

Label 2-600-600.461-c0-0-0
Degree $2$
Conductor $600$
Sign $0.876 + 0.481i$
Analytic cond. $0.299439$
Root an. cond. $0.547210$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.309 + 0.951i)6-s + 0.618·7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (1.30 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.500 + 0.363i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.309 + 0.951i)6-s + 0.618·7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (1.30 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.500 + 0.363i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.876 + 0.481i$
Analytic conductor: \(0.299439\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ 0.876 + 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6679621093\)
\(L(\frac12)\) \(\approx\) \(0.6679621093\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
good7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.97231207464096790647971221258, −9.732587778515541088125088080385, −8.611858368910480291177446067118, −8.235228171692228869539532436166, −7.32587980236027020902303822767, −6.57981879606240439600326884385, −5.82919538789292654622470160386, −4.16739019015564403310701070665, −2.71358618425737609668826823728, −1.16243574400570910889055909247, 1.67024574908609273310209588419, 3.31534862753941107268371344393, 4.21200835832611039258921885783, 4.95105270210484386958247208447, 6.79037239255847860279440445983, 7.79572856718561557423301554188, 8.595563872557490956263143927528, 9.179415868721389745919317888081, 9.968783006374190641432994872154, 10.92526713753748045312825538153

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