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Properties

Label 2-600-120.53-c1-0-5
Degree $2$
Conductor $600$
Sign $0.188 - 0.982i$
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  + (0.771 − 1.18i)2-s + (−1.57 + 0.713i)3-s + (−0.808 − 1.82i)4-s + (−0.373 + 2.42i)6-s + (−1.44 + 1.44i)7-s + (−2.79 − 0.454i)8-s + (1.98 − 2.25i)9-s − 0.641·11-s + (2.58 + 2.31i)12-s + (−2.03 + 2.03i)13-s + (0.597 + 2.83i)14-s + (−2.69 + 2.95i)16-s + (4.37 + 4.37i)17-s + (−1.13 − 4.08i)18-s − 4.93·19-s + ⋯
L(s)  = 1  + (0.545 − 0.837i)2-s + (−0.911 + 0.411i)3-s + (−0.404 − 0.914i)4-s + (−0.152 + 0.988i)6-s + (−0.546 + 0.546i)7-s + (−0.987 − 0.160i)8-s + (0.660 − 0.750i)9-s − 0.193·11-s + (0.744 + 0.667i)12-s + (−0.563 + 0.563i)13-s + (0.159 + 0.756i)14-s + (−0.673 + 0.739i)16-s + (1.06 + 1.06i)17-s + (−0.267 − 0.963i)18-s − 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.188 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445073 + 0.367666i\)
\(L(\frac12)\) \(\approx\) \(0.445073 + 0.367666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.771 + 1.18i)T \)
3 \( 1 + (1.57 - 0.713i)T \)
5 \( 1 \)
good7 \( 1 + (1.44 - 1.44i)T - 7iT^{2} \)
11 \( 1 + 0.641T + 11T^{2} \)
13 \( 1 + (2.03 - 2.03i)T - 13iT^{2} \)
17 \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \)
19 \( 1 + 4.93T + 19T^{2} \)
23 \( 1 + (3.73 - 3.73i)T - 23iT^{2} \)
29 \( 1 - 9.84iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + (3.44 + 3.44i)T + 37iT^{2} \)
41 \( 1 + 7.20iT - 41T^{2} \)
43 \( 1 + (4.37 - 4.37i)T - 43iT^{2} \)
47 \( 1 + (4.08 + 4.08i)T + 47iT^{2} \)
53 \( 1 + (-3.83 - 3.83i)T + 53iT^{2} \)
59 \( 1 + 2.50iT - 59T^{2} \)
61 \( 1 + 9.64iT - 61T^{2} \)
67 \( 1 + (2.59 + 2.59i)T + 67iT^{2} \)
71 \( 1 - 16.7iT - 71T^{2} \)
73 \( 1 + (8.40 + 8.40i)T + 73iT^{2} \)
79 \( 1 - 5.31iT - 79T^{2} \)
83 \( 1 + (2.20 + 2.20i)T + 83iT^{2} \)
89 \( 1 + 3.96T + 89T^{2} \)
97 \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \)
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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

−10.83478746230408094776951237469, −10.19799101682647889460088637294, −9.525533380659323299460205919317, −8.541023858048257016198524785545, −6.89791803943392404685316525026, −5.99311244191646133047035143579, −5.30589475873105529354759790793, −4.24115379586713043595402314889, −3.28288256450692956558901361760, −1.72057125151268574643665101513, 0.29405768950329859319788374102, 2.71776490086110342799025149008, 4.18269807328501849955502621033, 5.04448404961984274286647698798, 6.04381674756821773403312464817, 6.70135252324339408184859700385, 7.60949380560703573201328657646, 8.265917279082680062124293791324, 9.821250014426974563246550132630, 10.33753123292292676726051515354

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