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Properties

Label 2-600-15.2-c1-0-14
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: $\chi_{600} (257, \cdot )$
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

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

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