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Properties

Label 2-600-15.8-c1-0-4
Degree $2$
Conductor $600$
Sign $-0.161 - 0.986i$
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  + (0.0537 + 1.73i)3-s + (0.560 − 0.560i)7-s + (−2.99 + 0.186i)9-s − 0.939i·11-s + (4.13 + 4.13i)13-s + (4.50 + 4.50i)17-s + 3.74i·19-s + (0.999 + 0.939i)21-s + (−6.18 + 6.18i)23-s + (−0.483 − 5.17i)27-s − 3.16·29-s − 5.37·31-s + (1.62 − 0.0505i)33-s + (3.56 − 3.56i)37-s + (−6.92 + 7.37i)39-s + ⋯
L(s)  = 1  + (0.0310 + 0.999i)3-s + (0.211 − 0.211i)7-s + (−0.998 + 0.0620i)9-s − 0.283i·11-s + (1.14 + 1.14i)13-s + (1.09 + 1.09i)17-s + 0.859i·19-s + (0.218 + 0.205i)21-s + (−1.28 + 1.28i)23-s + (−0.0929 − 0.995i)27-s − 0.588·29-s − 0.964·31-s + (0.283 − 0.00879i)33-s + (0.586 − 0.586i)37-s + (−1.10 + 1.18i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.161 - 0.986i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.161 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915402 + 1.07735i\)
\(L(\frac12)\) \(\approx\) \(0.915402 + 1.07735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0537 - 1.73i)T \)
5 \( 1 \)
good7 \( 1 + (-0.560 + 0.560i)T - 7iT^{2} \)
11 \( 1 + 0.939iT - 11T^{2} \)
13 \( 1 + (-4.13 - 4.13i)T + 13iT^{2} \)
17 \( 1 + (-4.50 - 4.50i)T + 17iT^{2} \)
19 \( 1 - 3.74iT - 19T^{2} \)
23 \( 1 + (6.18 - 6.18i)T - 23iT^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + 5.37T + 31T^{2} \)
37 \( 1 + (-3.56 + 3.56i)T - 37iT^{2} \)
41 \( 1 + 9.15iT - 41T^{2} \)
43 \( 1 + (4.33 + 4.33i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (-0.526 + 0.526i)T - 53iT^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 1.37T + 61T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - 67iT^{2} \)
71 \( 1 - 3.16iT - 71T^{2} \)
73 \( 1 + (-5.56 - 5.56i)T + 73iT^{2} \)
79 \( 1 + 8.74iT - 79T^{2} \)
83 \( 1 + (-3.97 + 3.97i)T - 83iT^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (5.25 - 5.25i)T - 97iT^{2} \)
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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

−10.84152018670372631635415495747, −10.06985097505932974926212003798, −9.224084494336494414377291002361, −8.417814379618693984949800045219, −7.54147130355845717043500038619, −6.02562040549367982115445235135, −5.55043624214919765832641272243, −3.87107163067418606017083049105, −3.78036650193779792989608788342, −1.76718398759108403352845311040, 0.831066306431960825485182169127, 2.35062688380830638470589345961, 3.46375290604429486613754155440, 5.08291721385007521311702609002, 5.93849494189801647537539470241, 6.85928316167624074976648949071, 7.87988787539397777357960815582, 8.383192439312193034161767008836, 9.479253573336974251388754072807, 10.52897193386194381652763638398

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