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Properties

Label 2-600-25.21-c1-0-4
Degree $2$
Conductor $600$
Sign $0.0627 - 0.998i$
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  + (0.809 + 0.587i)3-s + (−1.80 + 1.31i)5-s + 3.23·7-s + (0.309 + 0.951i)9-s + (−1.38 + 4.25i)11-s + (−0.118 − 0.363i)13-s − 2.23·15-s + (−2.73 + 1.98i)17-s + (0.381 − 0.277i)19-s + (2.61 + 1.90i)21-s + (0.236 − 0.726i)23-s + (1.54 − 4.75i)25-s + (−0.309 + 0.951i)27-s + (7.35 + 5.34i)29-s + (−3.61 + 2.62i)31-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (−0.809 + 0.587i)5-s + 1.22·7-s + (0.103 + 0.317i)9-s + (−0.416 + 1.28i)11-s + (−0.0327 − 0.100i)13-s − 0.577·15-s + (−0.663 + 0.482i)17-s + (0.0876 − 0.0636i)19-s + (0.571 + 0.415i)21-s + (0.0492 − 0.151i)23-s + (0.309 − 0.951i)25-s + (−0.0594 + 0.183i)27-s + (1.36 + 0.992i)29-s + (−0.649 + 0.472i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.0627 - 0.998i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.0627 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11818 + 1.05004i\)
\(L(\frac12)\) \(\approx\) \(1.11818 + 1.05004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (1.80 - 1.31i)T \)
good7 \( 1 - 3.23T + 7T^{2} \)
11 \( 1 + (1.38 - 4.25i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.118 + 0.363i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.73 - 1.98i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.381 + 0.277i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.236 + 0.726i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-7.35 - 5.34i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.61 - 2.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.80 - 8.64i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.263 + 0.812i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.472T + 43T^{2} \)
47 \( 1 + (3 + 2.17i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-10.1 - 7.38i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.76 + 8.50i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.59 + 4.89i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (5.47 - 3.97i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (8.09 + 5.87i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.26 + 13.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.23 - 2.35i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.0 + 8.05i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-5.51 + 16.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.73 - 1.98i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.69067420723371124072187772138, −10.28076027090028296303638979333, −8.973855289098889784393335507754, −8.161150760193682104660207433145, −7.50943028294251091885201481833, −6.61511476708096758284658255357, −4.91817844593751014752802401278, −4.45075518853646126567384060502, −3.13786929326044591052477662733, −1.89504980343583476875765623258, 0.853653938400153197253198418532, 2.44232189988011591981591161472, 3.81760521244824432794914336692, 4.77187594647071316367179296323, 5.78849983774192550026830803170, 7.16479016369018474029044215766, 8.025953336645848316970046186519, 8.462505663595975924927978099579, 9.285119117835152751103057593962, 10.68905125600615006828983787813

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