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Properties

Label 2-600-600.581-c0-0-0
Degree $2$
Conductor $600$
Sign $-0.929 + 0.368i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s − 1.61·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.190 + 0.587i)11-s + (0.309 − 0.951i)12-s + (−0.500 − 1.53i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + 18-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s − 1.61·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.190 + 0.587i)11-s + (0.309 − 0.951i)12-s + (−0.500 − 1.53i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.929 + 0.368i$
Analytic conductor: \(0.299439\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ -0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5365102457\)
\(L(\frac12)\) \(\approx\) \(0.5365102457\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
good7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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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

−11.31950005221788735452879341167, −10.20274901874850271417652662893, −9.723669460073739776267157193982, −8.950068246658059596287880910925, −7.31852904345565715371835095778, −6.68869211954584936440997489726, −6.12378930611367341329444425505, −5.17359885429417921345660262352, −3.88265714929399562876545015626, −3.10719051896162680976184441992, 0.61723419447780696887501190023, 2.20860206905284398658715132605, 3.62819773786799494361511988085, 4.76479160159235241548278419525, 5.94778937317783289954358582731, 6.22063280972913537765628957527, 7.79478188794710945819909703724, 8.992187543682593437721113946648, 9.692900452249329745331970838407, 10.40226539939400498846125965897

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