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Properties

Label 2-600-120.59-c1-0-38
Degree $2$
Conductor $600$
Sign $-0.243 + 0.969i$
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 + 0.847i)2-s + (−1.71 − 0.242i)3-s + (0.562 − 1.91i)4-s + (2.14 − 1.17i)6-s − 3.08·7-s + (0.990 + 2.64i)8-s + (2.88 + 0.831i)9-s + 2.54i·11-s + (−1.42 + 3.15i)12-s + 5.06·13-s + (3.49 − 2.61i)14-s + (−3.36 − 2.15i)16-s − 0.214·17-s + (−3.96 + 1.50i)18-s − 2.60·19-s + ⋯
L(s)  = 1  + (−0.800 + 0.599i)2-s + (−0.990 − 0.139i)3-s + (0.281 − 0.959i)4-s + (0.876 − 0.481i)6-s − 1.16·7-s + (0.350 + 0.936i)8-s + (0.960 + 0.277i)9-s + 0.767i·11-s + (−0.412 + 0.910i)12-s + 1.40·13-s + (0.934 − 0.700i)14-s + (−0.841 − 0.539i)16-s − 0.0519·17-s + (−0.935 + 0.354i)18-s − 0.598·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.243 + 0.969i$
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: \((2,\ 600,\ (\ :1/2),\ -0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141939 - 0.182037i\)
\(L(\frac12)\) \(\approx\) \(0.141939 - 0.182037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.847i)T \)
3 \( 1 + (1.71 + 0.242i)T \)
5 \( 1 \)
good7 \( 1 + 3.08T + 7T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
13 \( 1 - 5.06T + 13T^{2} \)
17 \( 1 + 0.214T + 17T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + 4.58iT - 31T^{2} \)
37 \( 1 + 7.67T + 37T^{2} \)
41 \( 1 + 9.26iT - 41T^{2} \)
43 \( 1 - 11.4iT - 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 9.51iT - 53T^{2} \)
59 \( 1 - 0.428iT - 59T^{2} \)
61 \( 1 + 1.11iT - 61T^{2} \)
67 \( 1 + 2.35iT - 67T^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 - 2.29T + 83T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 - 8.04iT - 97T^{2} \)
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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

−10.38396872608762146076011417158, −9.587484276130257400607746438633, −8.734224339939524501908025670695, −7.57936150920474015971848821317, −6.63462753838885422792904148854, −6.22457425305496132465533985614, −5.21406952125556143227348705041, −3.89058202514979726055478454132, −1.88469877273354047624407649563, −0.20243571079126867607005771108, 1.35106823979007025552851233600, 3.24848437875122183110713522740, 3.99517588209696106170992288586, 5.69053895379659696992389454417, 6.43850936027529798588053925582, 7.30876435382513018963923519174, 8.580218349088698006472501861952, 9.310769324421700632670691794309, 10.18069954544464290584345343226, 10.93421754416470907540975704263

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