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Properties

Label 2-600-120.53-c1-0-37
Degree $2$
Conductor $600$
Sign $0.988 + 0.151i$
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  + (−1.21 − 0.722i)2-s + (0.889 + 1.48i)3-s + (0.956 + 1.75i)4-s + (−0.00827 − 2.44i)6-s + (2.09 − 2.09i)7-s + (0.106 − 2.82i)8-s + (−1.41 + 2.64i)9-s + 2.65·11-s + (−1.75 + 2.98i)12-s + (4.21 − 4.21i)13-s + (−4.05 + 1.03i)14-s + (−2.17 + 3.35i)16-s + (−3.84 − 3.84i)17-s + (3.63 − 2.19i)18-s + 3.15·19-s + ⋯
L(s)  = 1  + (−0.859 − 0.510i)2-s + (0.513 + 0.857i)3-s + (0.478 + 0.878i)4-s + (−0.00337 − 0.999i)6-s + (0.790 − 0.790i)7-s + (0.0377 − 0.999i)8-s + (−0.472 + 0.881i)9-s + 0.800·11-s + (−0.507 + 0.861i)12-s + (1.16 − 1.16i)13-s + (−1.08 + 0.275i)14-s + (−0.542 + 0.839i)16-s + (−0.933 − 0.933i)17-s + (0.856 − 0.516i)18-s + 0.724·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.988 + 0.151i$
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: \((2,\ 600,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34526 - 0.102753i\)
\(L(\frac12)\) \(\approx\) \(1.34526 - 0.102753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.722i)T \)
3 \( 1 + (-0.889 - 1.48i)T \)
5 \( 1 \)
good7 \( 1 + (-2.09 + 2.09i)T - 7iT^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + (-4.21 + 4.21i)T - 13iT^{2} \)
17 \( 1 + (3.84 + 3.84i)T + 17iT^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 + (1.60 - 1.60i)T - 23iT^{2} \)
29 \( 1 + 4.35iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (-4.94 - 4.94i)T + 37iT^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 + (-0.219 + 0.219i)T - 43iT^{2} \)
47 \( 1 + (-1.83 - 1.83i)T + 47iT^{2} \)
53 \( 1 + (-4.64 - 4.64i)T + 53iT^{2} \)
59 \( 1 + 9.93iT - 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 + (8.80 + 8.80i)T + 67iT^{2} \)
71 \( 1 + 2.89iT - 71T^{2} \)
73 \( 1 + (2.29 + 2.29i)T + 73iT^{2} \)
79 \( 1 + 9.02iT - 79T^{2} \)
83 \( 1 + (-9.78 - 9.78i)T + 83iT^{2} \)
89 \( 1 - 2.83T + 89T^{2} \)
97 \( 1 + (9.26 - 9.26i)T - 97iT^{2} \)
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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

−10.61034309130554691766526107962, −9.750442916439324679643367943500, −9.043475165603016059663457297629, −8.114165451766233397416679982729, −7.60538097731259219101927939721, −6.23898161598561716678796636100, −4.69619472438829811060456322038, −3.81398126868751573216768127715, −2.79846524902554607964759627276, −1.17731035557505737586030674185, 1.39807792133400836746255012275, 2.21510019314533504535942845634, 4.02087179784638136698578003456, 5.62617573428950777133736616565, 6.42932464849748955559605872774, 7.14934766548652212274241642535, 8.278205953178164022151612069261, 8.804029285096380962080948106768, 9.267891992828004324212098738609, 10.72618502697636886498961067506

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