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Properties

Label 2-600-24.11-c1-0-53
Degree $2$
Conductor $600$
Sign $0.333 + 0.942i$
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 + (−1.55 − 1.89i)6-s − 4.34i·7-s + (2.45 + 1.40i)8-s + (0.448 + 2.96i)9-s − 1.83i·11-s + (−1.70 − 3.01i)12-s + 0.588i·13-s + (1.06 − 6.05i)14-s + (3.07 + 2.55i)16-s − 5.37i·17-s + (−0.0995 + 4.24i)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 + (−0.634 − 0.773i)6-s − 1.64i·7-s + (0.867 + 0.497i)8-s + (0.149 + 0.988i)9-s − 0.553i·11-s + (−0.491 − 0.871i)12-s + 0.163i·13-s + (0.283 − 1.61i)14-s + (0.768 + 0.639i)16-s − 1.30i·17-s + (−0.0234 + 0.999i)18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.333 + 0.942i$
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: \((2,\ 600,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69777 - 1.20044i\)
\(L(\frac12)\) \(\approx\) \(1.69777 - 1.20044i\)
\(L(\frac{3}{2})\) 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

−10.84692774771285952887773020302, −10.05706145767294399880728413331, −8.330278712928439136051857680044, −7.39660535461251857141507405792, −6.82925425271750778946026979414, −6.03444131207776674155117499896, −4.82914792597864071561705104994, −4.14052501660676840948298825129, −2.67024557781757052972530825506, −0.972909857315987642003599393187, 1.97131419488216359021610439019, 3.25596577512938004787154871854, 4.47742850164835835319652936756, 5.23227165624493371870854980558, 6.10285581688048846385282803730, 6.68931611647861368400068533752, 8.310566419898740230320837663846, 9.222121298517881225578513587168, 10.36200976939857372968162664049, 10.84351664534942728116440398796

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