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Properties

Label 2-600-120.53-c1-0-32
Degree $2$
Conductor $600$
Sign $-0.0535 - 0.998i$
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  + (1.11 + 0.864i)2-s + (1.72 + 0.170i)3-s + (0.506 + 1.93i)4-s + (1.78 + 1.68i)6-s + (−2.06 + 2.06i)7-s + (−1.10 + 2.60i)8-s + (2.94 + 0.586i)9-s + 0.510·11-s + (0.543 + 3.42i)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 + (2.78 + 3.19i)18-s + 6.01·19-s + ⋯
L(s)  = 1  + (0.791 + 0.611i)2-s + (0.995 + 0.0982i)3-s + (0.253 + 0.967i)4-s + (0.727 + 0.685i)6-s + (−0.782 + 0.782i)7-s + (−0.390 + 0.920i)8-s + (0.980 + 0.195i)9-s + 0.153·11-s + (0.156 + 0.987i)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 + (0.656 + 0.754i)18-s + 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0535 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0535 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0535 - 0.998i$
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: \((2,\ 600,\ (\ :1/2),\ -0.0535 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99775 + 2.10771i\)
\(L(\frac12)\) \(\approx\) \(1.99775 + 2.10771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 - 0.864i)T \)
3 \( 1 + (-1.72 - 0.170i)T \)
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

−11.05140415449942842725997201738, −9.524468194115167359668881507638, −9.199328036113365348261766139977, −8.139606204292665286735791533241, −7.25511355907966489469093182382, −6.47072131063823693491672193225, −5.32624281519368892373883418154, −4.29276117426547481171816529463, −3.16384868538363832983077757301, −2.41462083059586532724528762257, 1.29158816133855631978692124515, 2.77208764036427958181885932215, 3.59660479731126588675491594891, 4.43813939210191656388669221253, 5.78405924465644881006630555211, 6.92360554293726842453309154216, 7.55409360244833499949366670986, 9.041124820292172047743691469576, 9.605514982444752647172689985533, 10.45842164470771637959021650809

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