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Properties

Label 2-600-15.2-c1-0-12
Degree $2$
Conductor $600$
Sign $0.994 + 0.0999i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.0999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.0999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.994 + 0.0999i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.994 + 0.0999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18861 - 0.109633i\)
\(L(\frac12)\) \(\approx\) \(2.18861 - 0.109633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.25 + 5.25i)T + 97iT^{2} \)
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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

−10.76362932623366140312083488042, −9.595831454933599269915263814049, −8.731942661831476445635394121750, −8.258209142913847969901614620294, −7.22354822670300121131876694586, −6.20365343415849044945680826373, −5.01520035661712834270095176149, −3.76734580627161032937780579094, −2.89913261518304885529090643788, −1.46093532894003912902420788738, 1.56226184924890598919430669106, 2.81232375876127428324526880289, 4.04901309822175519710316056738, 4.79636490253351934978825086653, 6.46337566511006500215380674817, 7.10423011887002433218972538114, 8.210270072033820665541006423041, 8.934003294867537557318300818792, 9.561671755781871117392239089790, 10.69772425175893871180534418244

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