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Properties

Label 2-600-120.59-c1-0-8
Degree $2$
Conductor $600$
Sign $-0.957 - 0.287i$
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  + (−1.29 + 0.578i)2-s + (1.56 + 0.751i)3-s + (1.33 − 1.49i)4-s + (−2.44 − 0.0670i)6-s − 4.28·7-s + (−0.852 + 2.69i)8-s + (1.86 + 2.34i)9-s + 2.44i·11-s + (3.19 − 1.33i)12-s − 2.71·13-s + (5.53 − 2.48i)14-s + (−0.460 − 3.97i)16-s − 1.16·17-s + (−3.77 − 1.94i)18-s − 6.05·19-s + ⋯
L(s)  = 1  + (−0.912 + 0.409i)2-s + (0.900 + 0.433i)3-s + (0.665 − 0.746i)4-s + (−0.999 − 0.0273i)6-s − 1.61·7-s + (−0.301 + 0.953i)8-s + (0.623 + 0.781i)9-s + 0.737i·11-s + (0.923 − 0.384i)12-s − 0.752·13-s + (1.47 − 0.662i)14-s + (−0.115 − 0.993i)16-s − 0.282·17-s + (−0.888 − 0.458i)18-s − 1.38·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.957 - 0.287i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.957 - 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0875968 + 0.596854i\)
\(L(\frac12)\) \(\approx\) \(0.0875968 + 0.596854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.578i)T \)
3 \( 1 + (-1.56 - 0.751i)T \)
5 \( 1 \)
good7 \( 1 + 4.28T + 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + 2.71T + 13T^{2} \)
17 \( 1 + 1.16T + 17T^{2} \)
19 \( 1 + 6.05T + 19T^{2} \)
23 \( 1 - 7.55iT - 23T^{2} \)
29 \( 1 - 0.733T + 29T^{2} \)
31 \( 1 + 0.469iT - 31T^{2} \)
37 \( 1 - 1.36T + 37T^{2} \)
41 \( 1 - 4.69iT - 41T^{2} \)
43 \( 1 + 1.50iT - 43T^{2} \)
47 \( 1 + 4.07iT - 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 - 1.63iT - 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 - 9.97iT - 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 9.63iT - 73T^{2} \)
79 \( 1 + 3.61iT - 79T^{2} \)
83 \( 1 - 5.45T + 83T^{2} \)
89 \( 1 + 7.75iT - 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−10.46897706050494130471873257268, −9.957337537449922551353709561275, −9.351408450009912317887783639937, −8.653900820001217836079487363464, −7.50269052976450087247768712333, −6.92089538008185016891447663923, −5.85541217082476968548321769595, −4.47063904811852463814319403579, −3.13525943865754249489629151183, −2.07571312863065881747715948903, 0.36871250568059751031547253042, 2.30650396967709347763536781537, 3.06110793551202221376929465518, 4.10600351902360038196798103553, 6.36788369601848949327888102074, 6.69141746791956825029029936322, 7.83891308407131575285699040576, 8.700663899116675339700396333603, 9.284918847072507174975964883029, 10.10972798727162364425307993109

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