Properties

Label 2-600-8.5-c1-0-22
Degree $2$
Conductor $600$
Sign $0.951 - 0.306i$
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  + (1.29 + 0.576i)2-s i·3-s + (1.33 + 1.48i)4-s + (0.576 − 1.29i)6-s + 1.97·7-s + (0.867 + 2.69i)8-s − 9-s − 1.43i·11-s + (1.48 − 1.33i)12-s − 0.241i·13-s + (2.55 + 1.13i)14-s + (−0.430 + 3.97i)16-s + 7.38·17-s + (−1.29 − 0.576i)18-s + 3.04i·19-s + ⋯
L(s)  = 1  + (0.913 + 0.407i)2-s − 0.577i·3-s + (0.667 + 0.744i)4-s + (0.235 − 0.527i)6-s + 0.747·7-s + (0.306 + 0.951i)8-s − 0.333·9-s − 0.431i·11-s + (0.429 − 0.385i)12-s − 0.0669i·13-s + (0.682 + 0.304i)14-s + (−0.107 + 0.994i)16-s + 1.79·17-s + (−0.304 − 0.135i)18-s + 0.697i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.951 - 0.306i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70919 + 0.425837i\)
\(L(\frac12)\) \(\approx\) \(2.70919 + 0.425837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.576i)T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - 1.97T + 7T^{2} \)
11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 + 0.241iT - 13T^{2} \)
17 \( 1 - 7.38T + 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 + 0.874T + 23T^{2} \)
29 \( 1 + 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 - 8.81iT - 37T^{2} \)
41 \( 1 + 1.91T + 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 - 3.34T + 47T^{2} \)
53 \( 1 - 9.20iT - 53T^{2} \)
59 \( 1 + 6.43iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 - 4.86iT - 67T^{2} \)
71 \( 1 + 8.21T + 71T^{2} \)
73 \( 1 - 4.12T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 12.3iT - 83T^{2} \)
89 \( 1 + 8.08T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−11.00392052778941069711791334951, −9.969600843357573532879450366896, −8.488553424382952677387906281674, −7.88709990641552762827197711626, −7.17018283706080543878767299439, −5.91955173920481645116687039909, −5.44168921213035579537916499124, −4.13144826009036320025101826925, −3.04679607102503896855666445400, −1.63916164355604424296735014084, 1.54742336794360798497906840171, 3.00509348355200390685138954394, 4.00445723364029021260715701800, 5.05686014890892978787199227387, 5.59580870556383367900069313382, 6.94944112096665243952875224026, 7.84056499707213385738348136824, 9.144084371484516344277340466532, 9.956969796536183313650350395953, 10.82274591911613687981245623746

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