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Properties

Label 2-600-120.53-c1-0-40
Degree $2$
Conductor $600$
Sign $0.507 + 0.861i$
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 + (−0.713 + 1.57i)3-s + (−0.808 − 1.82i)4-s + (−1.31 − 2.06i)6-s + (−1.44 + 1.44i)7-s + (2.79 + 0.454i)8-s + (−1.98 − 2.25i)9-s + 0.641·11-s + (3.46 + 0.0287i)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 + (4.19 − 0.612i)18-s − 4.93·19-s + ⋯
L(s)  = 1  + (−0.545 + 0.837i)2-s + (−0.411 + 0.911i)3-s + (−0.404 − 0.914i)4-s + (−0.538 − 0.842i)6-s + (−0.546 + 0.546i)7-s + (0.987 + 0.160i)8-s + (−0.660 − 0.750i)9-s + 0.193·11-s + (0.999 + 0.00829i)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 + (0.989 − 0.144i)18-s − 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.507 + 0.861i$
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: \((2,\ 600,\ (\ :1/2),\ 0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.137410 - 0.0785451i\)
\(L(\frac12)\) \(\approx\) \(0.137410 - 0.0785451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.713 - 1.57i)T \)
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

−10.30674554809821480642799747867, −9.356046588887254892754530623062, −9.104330637628505343822795086978, −7.993229781303495656313769616900, −6.55388830841940632664145985665, −6.30134810816234496826891155763, −4.92111835753608991318523857175, −4.36135322352935895505767138922, −2.56170788063191771031802101205, −0.11211610008010164049225075841, 1.45473542082663568494663325527, 2.71206744342077708702657511121, 3.96124556147204592058477590685, 5.24985719012355884780769717252, 6.68865735522347752585882187300, 7.18411514471840822515091265448, 8.360686349770297341457234031148, 8.941053418534780105364666882056, 10.38359768626863831478606429934, 10.60431765807236582758145196883

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