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Properties

Label 2-600-120.59-c1-0-51
Degree $2$
Conductor $600$
Sign $0.881 - 0.472i$
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  + 1.41·2-s + (1.41 + i)3-s + 2.00·4-s + (2.00 + 1.41i)6-s + 2.82·8-s + (1.00 + 2.82i)9-s − 2.82i·11-s + (2.82 + 2.00i)12-s + 4.00·16-s − 5.65·17-s + (1.41 + 4.00i)18-s − 2·19-s − 4.00i·22-s + (4.00 + 2.82i)24-s + (−1.41 + 5.00i)27-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.816 + 0.577i)3-s + 1.00·4-s + (0.816 + 0.577i)6-s + 1.00·8-s + (0.333 + 0.942i)9-s − 0.852i·11-s + (0.816 + 0.577i)12-s + 1.00·16-s − 1.37·17-s + (0.333 + 0.942i)18-s − 0.458·19-s − 0.852i·22-s + (0.816 + 0.577i)24-s + (−0.272 + 0.962i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.37086 + 0.845781i\)
\(L(\frac12)\) \(\approx\) \(3.37086 + 0.845781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (-1.41 - i)T \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.92737328432272080191776654319, −9.997601080926323040730778724661, −8.889001917733074966043593752805, −8.136759493278459585253749497716, −7.05863794170399786235081727344, −6.07043984943502672676692237312, −4.93792359669586807166752479724, −4.08835243987804237648562223891, −3.12964774090106058617758216223, −2.06099318866161002582582951377, 1.78062738312722353267386411657, 2.71243420627263156625064686727, 3.95067493226425855498564030044, 4.80767819429344485435476403478, 6.23869135863595461806404033121, 6.90146550392018586097333684497, 7.73218109799952271377550236710, 8.711600663394994580389208278670, 9.739283702727027508859370367069, 10.75151571126252877408356417297

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