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Properties

Label 2-600-24.11-c1-0-48
Degree $2$
Conductor $600$
Sign $-0.982 + 0.188i$
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  + (−1.39 − 0.244i)2-s + (−1.31 + 1.12i)3-s + (1.88 + 0.680i)4-s + (2.10 − 1.25i)6-s − 4.34i·7-s + (−2.45 − 1.40i)8-s + (0.448 − 2.96i)9-s + 1.83i·11-s + (−3.23 + 1.23i)12-s + 0.588i·13-s + (−1.06 + 6.05i)14-s + (3.07 + 2.55i)16-s + 5.37i·17-s + (−1.34 + 4.02i)18-s − 5.38·19-s + ⋯
L(s)  = 1  + (−0.984 − 0.172i)2-s + (−0.758 + 0.652i)3-s + (0.940 + 0.340i)4-s + (0.859 − 0.511i)6-s − 1.64i·7-s + (−0.867 − 0.497i)8-s + (0.149 − 0.988i)9-s + 0.553i·11-s + (−0.934 + 0.355i)12-s + 0.163i·13-s + (−0.283 + 1.61i)14-s + (0.768 + 0.639i)16-s + 1.30i·17-s + (−0.317 + 0.948i)18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.982 + 0.188i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00693617 - 0.0728545i\)
\(L(\frac12)\) \(\approx\) \(0.00693617 - 0.0728545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.244i)T \)
3 \( 1 + (1.31 - 1.12i)T \)
5 \( 1 \)
good7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 - 1.83iT - 11T^{2} \)
13 \( 1 - 0.588iT - 13T^{2} \)
17 \( 1 - 5.37iT - 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + 2.40T + 23T^{2} \)
29 \( 1 + 7.98T + 29T^{2} \)
31 \( 1 + 7.06iT - 31T^{2} \)
37 \( 1 - 2.72iT - 37T^{2} \)
41 \( 1 + 3.42iT - 41T^{2} \)
43 \( 1 + 2.96T + 43T^{2} \)
47 \( 1 + 9.81T + 47T^{2} \)
53 \( 1 - 6.65T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 9.27iT - 61T^{2} \)
67 \( 1 + 4.13T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 4.42T + 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 - 4.21iT - 89T^{2} \)
97 \( 1 + 2.16T + 97T^{2} \)
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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

−10.21246449892681883574210199085, −9.772054522375539621588171401680, −8.598131038121016749259886471677, −7.59064631060436923294736690328, −6.76312330761135341774028159724, −5.94482712612773211135051580079, −4.31538733485520266084197390934, −3.71709605481746941881348123443, −1.68958963855640806490396489584, −0.05919363940459307048160349351, 1.80328622107225989924124052729, 2.82436525113404361556022604229, 5.11786183029043043595402076264, 5.84869821007531539390829237921, 6.57980704940767657351477655383, 7.59423071728037699586871395577, 8.520734901225416002909768403656, 9.127276044397856501199648659722, 10.21622833503050956962501182380, 11.17545286656483485363843064384

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