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Properties

Label 2-600-120.59-c1-0-5
Degree $2$
Conductor $600$
Sign $-0.671 - 0.741i$
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  + (−0.639 − 1.26i)2-s + (0.730 + 1.57i)3-s + (−1.18 + 1.61i)4-s + (1.51 − 1.92i)6-s − 1.25·7-s + (2.79 + 0.458i)8-s + (−1.93 + 2.29i)9-s + 3.02i·11-s + (−3.39 − 0.677i)12-s − 5.65·13-s + (0.803 + 1.58i)14-s + (−1.20 − 3.81i)16-s − 2.45·17-s + (4.12 + 0.972i)18-s − 1.77·19-s + ⋯
L(s)  = 1  + (−0.452 − 0.891i)2-s + (0.421 + 0.906i)3-s + (−0.590 + 0.806i)4-s + (0.618 − 0.786i)6-s − 0.474·7-s + (0.986 + 0.162i)8-s + (−0.644 + 0.764i)9-s + 0.911i·11-s + (−0.980 − 0.195i)12-s − 1.56·13-s + (0.214 + 0.423i)14-s + (−0.301 − 0.953i)16-s − 0.595·17-s + (0.973 + 0.229i)18-s − 0.407·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.671 - 0.741i$
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: \((2,\ 600,\ (\ :1/2),\ -0.671 - 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189656 + 0.427688i\)
\(L(\frac12)\) \(\approx\) \(0.189656 + 0.427688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.639 + 1.26i)T \)
3 \( 1 + (-0.730 - 1.57i)T \)
5 \( 1 \)
good7 \( 1 + 1.25T + 7T^{2} \)
11 \( 1 - 3.02iT - 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 2.45T + 17T^{2} \)
19 \( 1 + 1.77T + 19T^{2} \)
23 \( 1 + 8.84iT - 23T^{2} \)
29 \( 1 - 3.79T + 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + 6.45T + 37T^{2} \)
41 \( 1 - 7.57iT - 41T^{2} \)
43 \( 1 - 4.37iT - 43T^{2} \)
47 \( 1 + 1.83iT - 47T^{2} \)
53 \( 1 - 12.0iT - 53T^{2} \)
59 \( 1 + 4.91iT - 59T^{2} \)
61 \( 1 - 8.16iT - 61T^{2} \)
67 \( 1 - 8.50iT - 67T^{2} \)
71 \( 1 + 7.00T + 71T^{2} \)
73 \( 1 + 4.59iT - 73T^{2} \)
79 \( 1 + 7.36iT - 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−10.48363273795763907825676209704, −10.28890619565260700027460080313, −9.381604549772118678596683805528, −8.726996013213213877690893313648, −7.74671419953978963201157848323, −6.68961499513885104874951883007, −4.81707324229454554338070694997, −4.46054530996416557327071339248, −3.02926139979114722850278581535, −2.24650322682217920778351598786, 0.26818772155468267460047516047, 2.05003184398638124715203761305, 3.53949230246622903468399259586, 5.11251963046798528506491364031, 6.06211610565401262313232769038, 6.93139493343396875769974337370, 7.58554543355833019709747990787, 8.464303742056594341361141461574, 9.260744234829917040051793244343, 9.981670132704469211372100516514

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