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Properties

Label 2-600-15.8-c1-0-0
Degree $2$
Conductor $600$
Sign $-0.998 - 0.0618i$
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  + (0.292 + 1.70i)3-s + (−3 + 3i)7-s + (−2.82 + i)9-s − 1.41i·11-s + (−4.24 − 4.24i)17-s + 4i·19-s + (−5.99 − 4.24i)21-s + (−2.82 + 2.82i)23-s + (−2.53 − 4.53i)27-s + 1.41·29-s − 2·31-s + (2.41 − 0.414i)33-s + (2 − 2i)37-s + 5.65i·41-s + (2 + 2i)43-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + (−1.13 + 1.13i)7-s + (−0.942 + 0.333i)9-s − 0.426i·11-s + (−1.02 − 1.02i)17-s + 0.917i·19-s + (−1.30 − 0.925i)21-s + (−0.589 + 0.589i)23-s + (−0.487 − 0.872i)27-s + 0.262·29-s − 0.359·31-s + (0.420 − 0.0721i)33-s + (0.328 − 0.328i)37-s + 0.883i·41-s + (0.304 + 0.304i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.998 - 0.0618i$
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: \((2,\ 600,\ (\ :1/2),\ -0.998 - 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0221385 + 0.715011i\)
\(L(\frac12)\) \(\approx\) \(0.0221385 + 0.715011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 \)
good7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-2 + 2i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (8.48 - 8.48i)T - 53iT^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (-13 + 13i)T - 97iT^{2} \)
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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

−11.05102373354392364033596380102, −10.02550950235950466064574609109, −9.318439507832295538755595586554, −8.839222218878613127584841948909, −7.71282006566400419265441016062, −6.26721286446570257336003727360, −5.70209234447232390769790061784, −4.52781797525554858933349514616, −3.35316540524734806991000923060, −2.51149917893159982814059512944, 0.36374035517925994833875715008, 2.06320607704949247415784799678, 3.36461889612625722427737669008, 4.44086421624489791051725399523, 6.04126992521340055942578711387, 6.75359766451144826774473286639, 7.34290803399957647415589062383, 8.424331975250764350954235776546, 9.291332962565328116047975399033, 10.31727835768580726626022715833

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