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Properties

Label 2-600-120.59-c1-0-7
Degree $2$
Conductor $600$
Sign $0.777 - 0.628i$
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  + (−0.842 − 1.13i)2-s + (0.218 − 1.71i)3-s + (−0.581 + 1.91i)4-s + (−2.13 + 1.19i)6-s − 3.64·7-s + (2.66 − 0.949i)8-s + (−2.90 − 0.750i)9-s + 5.07i·11-s + (3.16 + 1.41i)12-s − 1.70·13-s + (3.06 + 4.14i)14-s + (−3.32 − 2.22i)16-s + 4.08·17-s + (1.59 + 3.93i)18-s + 1.26·19-s + ⋯
L(s)  = 1  + (−0.595 − 0.803i)2-s + (0.126 − 0.992i)3-s + (−0.290 + 0.956i)4-s + (−0.872 + 0.489i)6-s − 1.37·7-s + (0.941 − 0.335i)8-s + (−0.968 − 0.250i)9-s + 1.52i·11-s + (0.912 + 0.409i)12-s − 0.473·13-s + (0.820 + 1.10i)14-s + (−0.830 − 0.556i)16-s + 0.989·17-s + (0.375 + 0.926i)18-s + 0.290·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.777 - 0.628i$
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: \((2,\ 600,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417049 + 0.147450i\)
\(L(\frac12)\) \(\approx\) \(0.417049 + 0.147450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.842 + 1.13i)T \)
3 \( 1 + (-0.218 + 1.71i)T \)
5 \( 1 \)
good7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 - 5.07iT - 11T^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
17 \( 1 - 4.08T + 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 - 4.70iT - 23T^{2} \)
29 \( 1 + 1.06T + 29T^{2} \)
31 \( 1 - 4.86iT - 31T^{2} \)
37 \( 1 + 7.56T + 37T^{2} \)
41 \( 1 + 1.50iT - 41T^{2} \)
43 \( 1 - 3.43iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + 8.87iT - 53T^{2} \)
59 \( 1 - 0.788iT - 59T^{2} \)
61 \( 1 - 0.627iT - 61T^{2} \)
67 \( 1 + 4.18iT - 67T^{2} \)
71 \( 1 - 6.21T + 71T^{2} \)
73 \( 1 + 4.21iT - 73T^{2} \)
79 \( 1 - 0.992iT - 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 - 7.40iT - 97T^{2} \)
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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

−10.65496914014894276432141722590, −9.646020883017477927993758302229, −9.398145860361505368662234096507, −8.063005476836436181457973089664, −7.28349803318278753821979831518, −6.68165655919363170444942584589, −5.22860292695843515555645558337, −3.63372159020176067020014951054, −2.73467940180429537949133914303, −1.50044227595646367225993720138, 0.29773169691891100667852347425, 2.90780663601229643980236713657, 3.92701378235565574133412618370, 5.36394814031758363988382540348, 5.94745270504568842721123423398, 6.94461937612150624842976816947, 8.167830332164048441560729613041, 8.859776690351192540791394247373, 9.675365278347775767055742828146, 10.23608597452920294803749847026

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