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Properties

Label 2-600-600.269-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  + (−0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (0.809 + 0.587i)6-s + 1.17i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.587 + 1.80i)11-s + (−0.951 − 0.309i)12-s + (−0.363 − 1.11i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s − i·18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (0.809 + 0.587i)6-s + 1.17i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.587 + 1.80i)11-s + (−0.951 − 0.309i)12-s + (−0.363 − 1.11i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s − i·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: $\chi_{600} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5948365569\)
\(L(\frac12)\) \(\approx\) \(0.5948365569\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
good7 \( 1 - 1.17iT - T^{2} \)
11 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
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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

−10.84935404927435035553285120353, −9.766197470006057211227399264327, −9.303650720424895906397400455737, −8.328996639223767582495123732355, −7.28990616927723640130521004273, −6.52826458645548242578845104036, −5.72287815479289663748488257850, −4.94694441963413401422875354919, −2.30175388911260252565641256415, −1.73185688371429271339052146569, 1.12906803275842967579368664750, 3.09288458962491938178322372321, 3.93007632761918969366830900123, 5.55901385499399770073448758485, 6.37217345442334039029677051903, 7.18913176712008091054108553158, 8.574629719688482946526687489112, 9.234593107131618891646345542685, 10.08444524678933138104431287326, 10.85037737361678941775573257255

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