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Properties

Label 2-600-120.53-c1-0-3
Degree $2$
Conductor $600$
Sign $-0.869 - 0.494i$
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  + (−0.864 − 1.11i)2-s + (0.170 + 1.72i)3-s + (−0.506 + 1.93i)4-s + (1.78 − 1.68i)6-s + (−2.06 + 2.06i)7-s + (2.60 − 1.10i)8-s + (−2.94 + 0.586i)9-s + 0.510·11-s + (−3.42 − 0.543i)12-s + (0.750 − 0.750i)13-s + (4.10 + 0.528i)14-s + (−3.48 − 1.95i)16-s + (3.14 + 3.14i)17-s + (3.19 + 2.78i)18-s − 6.01·19-s + ⋯
L(s)  = 1  + (−0.611 − 0.791i)2-s + (0.0982 + 0.995i)3-s + (−0.253 + 0.967i)4-s + (0.727 − 0.685i)6-s + (−0.782 + 0.782i)7-s + (0.920 − 0.390i)8-s + (−0.980 + 0.195i)9-s + 0.153·11-s + (−0.987 − 0.156i)12-s + (0.208 − 0.208i)13-s + (1.09 + 0.141i)14-s + (−0.871 − 0.489i)16-s + (0.763 + 0.763i)17-s + (0.754 + 0.656i)18-s − 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.869 - 0.494i$
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: \((2,\ 600,\ (\ :1/2),\ -0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107319 + 0.405708i\)
\(L(\frac12)\) \(\approx\) \(0.107319 + 0.405708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.864 + 1.11i)T \)
3 \( 1 + (-0.170 - 1.72i)T \)
5 \( 1 \)
good7 \( 1 + (2.06 - 2.06i)T - 7iT^{2} \)
11 \( 1 - 0.510T + 11T^{2} \)
13 \( 1 + (-0.750 + 0.750i)T - 13iT^{2} \)
17 \( 1 + (-3.14 - 3.14i)T + 17iT^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + (2.54 - 2.54i)T - 23iT^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (6.76 + 6.76i)T + 37iT^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + (5.95 - 5.95i)T - 43iT^{2} \)
47 \( 1 + (3.33 + 3.33i)T + 47iT^{2} \)
53 \( 1 + (5.75 + 5.75i)T + 53iT^{2} \)
59 \( 1 + 1.16iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + (-7.98 - 7.98i)T + 67iT^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 + (-3.20 - 3.20i)T + 73iT^{2} \)
79 \( 1 - 7.31iT - 79T^{2} \)
83 \( 1 + (-4.77 - 4.77i)T + 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (10.8 - 10.8i)T - 97iT^{2} \)
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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

−10.85888156078278826282377661927, −10.04073242880731034322895168847, −9.506088503575885223011941692442, −8.628933990343791398659566151723, −8.014285978307210204000228680497, −6.46028097655810775424900604801, −5.43689757133857116674449848616, −4.06067070395631170967499625161, −3.34239711764647001386323530477, −2.14775524715848437736168180436, 0.27134231242564449299341591822, 1.77752323869826850143376437867, 3.47902438510360410019587315503, 5.00026196340612408417105987176, 6.23363586091121440771021506818, 6.76537692988724493421791811153, 7.51047695807222871361150321101, 8.431669939966809666903062827757, 9.187896062058714993334464168245, 10.21149706950431482330092389750

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