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Properties

Label 2-600-5.4-c3-0-19
Degree $2$
Conductor $600$
Sign $-0.447 + 0.894i$
Analytic cond. $35.4011$
Root an. cond. $5.94988$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 20i·7-s − 9·9-s − 56·11-s + 86i·13-s − 106i·17-s − 4·19-s − 60·21-s − 136i·23-s − 27i·27-s + 206·29-s − 152·31-s − 168i·33-s + 282i·37-s − 258·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.07i·7-s − 0.333·9-s − 1.53·11-s + 1.83i·13-s − 1.51i·17-s − 0.0482·19-s − 0.623·21-s − 1.23i·23-s − 0.192i·27-s + 1.31·29-s − 0.880·31-s − 0.886i·33-s + 1.25i·37-s − 1.05·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(35.4011\)
Root analytic conductor: \(5.94988\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 600,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 - 20iT - 343T^{2} \)
11 \( 1 + 56T + 1.33e3T^{2} \)
13 \( 1 - 86iT - 2.19e3T^{2} \)
17 \( 1 + 106iT - 4.91e3T^{2} \)
19 \( 1 + 4T + 6.85e3T^{2} \)
23 \( 1 + 136iT - 1.21e4T^{2} \)
29 \( 1 - 206T + 2.43e4T^{2} \)
31 \( 1 + 152T + 2.97e4T^{2} \)
37 \( 1 - 282iT - 5.06e4T^{2} \)
41 \( 1 + 246T + 6.89e4T^{2} \)
43 \( 1 + 412iT - 7.95e4T^{2} \)
47 \( 1 - 40iT - 1.03e5T^{2} \)
53 \( 1 - 126iT - 1.48e5T^{2} \)
59 \( 1 + 56T + 2.05e5T^{2} \)
61 \( 1 + 2T + 2.26e5T^{2} \)
67 \( 1 + 388iT - 3.00e5T^{2} \)
71 \( 1 + 672T + 3.57e5T^{2} \)
73 \( 1 + 1.17e3iT - 3.89e5T^{2} \)
79 \( 1 + 408T + 4.93e5T^{2} \)
83 \( 1 + 668iT - 5.71e5T^{2} \)
89 \( 1 + 66T + 7.04e5T^{2} \)
97 \( 1 + 926iT - 9.12e5T^{2} \)
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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

−9.961330259366870301714752079040, −9.034437486085000684684456868724, −8.522105176246537065099730048452, −7.27567543719548659570757669931, −6.28594904743092998073887404083, −5.12500783878217281851563795407, −4.59538274183147697469467042266, −2.99618877926623599753681107198, −2.17341438147151103132788865928, 0, 1.23501937385760517758318041152, 2.74502401999040569148335862030, 3.77651573908943329558227462050, 5.20234415738391452486477451965, 5.93299351961544439853575582028, 7.21287775648466853315084680363, 7.86197692270884753898154690220, 8.420965240853881810552966229208, 10.02625051401963210467891274654

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