Properties

Label 2-600-600.509-c0-0-1
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  + (−0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.309 + 0.951i)6-s + 1.90i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.951 − 0.690i)11-s + (−0.587 − 0.809i)12-s + (−1.53 − 1.11i)14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + 0.999i·18-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.309 + 0.951i)6-s + 1.90i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.951 − 0.690i)11-s + (−0.587 − 0.809i)12-s + (−1.53 − 1.11i)14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + 0.999i·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: $\chi_{600} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9633640091\)
\(L(\frac12)\) \(\approx\) \(0.9633640091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
good7 \( 1 - 1.90iT - T^{2} \)
11 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
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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

−10.60360485254817116202178101806, −9.525339031429001522540908574033, −9.026065809700060062262359292410, −8.419936306732642812229206997411, −7.79135255341540317381967523891, −6.36674009833412692200171429904, −5.65387377422654903464015565908, −4.81707902390565063278476739748, −2.83088175948821653773864595132, −1.75462904858523276402344065697, 1.75994906341256531094392674711, 2.92727719967910407026117076755, 3.84014914999646046432344837724, 4.78093643028151439514457660078, 6.82978531837306627117158375433, 7.51452233090596355162791455692, 8.094143639971374493612283476839, 9.533149096865469411332642290373, 9.920647053838673701371158469957, 10.65178954520270775939815658743

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