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Properties

Label 2-600-24.11-c1-0-59
Degree $2$
Conductor $600$
Sign $0.451 + 0.892i$
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  + (1.13 − 0.842i)2-s + (1.71 − 0.218i)3-s + (0.581 − 1.91i)4-s + (1.76 − 1.69i)6-s + 3.64i·7-s + (−0.949 − 2.66i)8-s + (2.90 − 0.750i)9-s − 5.07i·11-s + (0.581 − 3.41i)12-s − 1.70i·13-s + (3.06 + 4.14i)14-s + (−3.32 − 2.22i)16-s + 4.08i·17-s + (2.66 − 3.29i)18-s − 1.26·19-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)2-s + (0.992 − 0.126i)3-s + (0.290 − 0.956i)4-s + (0.721 − 0.691i)6-s + 1.37i·7-s + (−0.335 − 0.941i)8-s + (0.968 − 0.250i)9-s − 1.52i·11-s + (0.168 − 0.985i)12-s − 0.473i·13-s + (0.820 + 1.10i)14-s + (−0.830 − 0.556i)16-s + 0.989i·17-s + (0.628 − 0.777i)18-s − 0.290·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.451 + 0.892i$
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: \((2,\ 600,\ (\ :1/2),\ 0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66745 - 1.63881i\)
\(L(\frac12)\) \(\approx\) \(2.66745 - 1.63881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.842i)T \)
3 \( 1 + (-1.71 + 0.218i)T \)
5 \( 1 \)
good7 \( 1 - 3.64iT - 7T^{2} \)
11 \( 1 + 5.07iT - 11T^{2} \)
13 \( 1 + 1.70iT - 13T^{2} \)
17 \( 1 - 4.08iT - 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 + 1.06T + 29T^{2} \)
31 \( 1 - 4.86iT - 31T^{2} \)
37 \( 1 - 7.56iT - 37T^{2} \)
41 \( 1 - 1.50iT - 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 8.87T + 53T^{2} \)
59 \( 1 - 0.788iT - 59T^{2} \)
61 \( 1 - 0.627iT - 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 + 6.21T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 + 0.992iT - 79T^{2} \)
83 \( 1 - 7.72iT - 83T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 - 7.40T + 97T^{2} \)
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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

−10.64262318382475745124989930199, −9.636567559720870551630462236875, −8.719256091415095671009110745722, −8.218395340561229273986421132928, −6.63533232785492082165577815549, −5.85620139255038045939870930987, −4.84034374861317670728737640645, −3.38246235596384214370231708786, −2.89473228736848096202144601013, −1.56329417851536531035883420773, 2.02476146950355173636639211254, 3.34691056002466907281241085458, 4.34051718965530342063059867593, 4.85956075569198749433578707987, 6.64492286333399776743287011637, 7.30692517432981378652905086081, 7.74923025413992279737183138692, 9.040043033249541918166528294608, 9.789002248611868214280104030491, 10.79876328691613993784979963654

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