↑

Properties

Label 2-600-24.11-c1-0-21
Degree $2$
Conductor $600$
Sign $-0.918 - 0.395i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 + 0.866i)2-s + 1.73i·3-s + (0.500 + 1.93i)4-s + (−1.49 + 1.93i)6-s + (−1.11 + 2.59i)8-s − 2.99·9-s + (−3.35 + 0.866i)12-s + (−3.5 + 1.93i)16-s + 6.92i·17-s + (−3.35 − 2.59i)18-s − 4·19-s + 8.94·23-s + (−4.50 − 1.93i)24-s − 5.19i·27-s − 7.74i·31-s + (−5.59 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)2-s + 0.999i·3-s + (0.250 + 0.968i)4-s + (−0.612 + 0.790i)6-s + (−0.395 + 0.918i)8-s − 0.999·9-s + (−0.968 + 0.250i)12-s + (−0.875 + 0.484i)16-s + 1.68i·17-s + (−0.790 − 0.612i)18-s − 0.917·19-s + 1.86·23-s + (−0.918 − 0.395i)24-s − 0.999i·27-s − 1.39i·31-s + (−0.988 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.918 - 0.395i$
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: \((2,\ 600,\ (\ :1/2),\ -0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407886 + 1.97972i\)
\(L(\frac12)\) \(\approx\) \(0.407886 + 1.97972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 - 0.866i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 8.94T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.98111191692415445951788246213, −10.38431256208444974439546706192, −9.032149469823094142667806816561, −8.515381620005814850066805656850, −7.41583346867162781325024576245, −6.24568130251168253340295916144, −5.54770102164426611730704694018, −4.43884623727815658805114866687, −3.77864372974204266585020853295, −2.54206740467330219094156722936, 0.891855074449704594312814741654, 2.34633034388162278141777002583, 3.23855542968127472546251401678, 4.75978776252473439105579639596, 5.56007435356991714005139779570, 6.75215948023137960166942279914, 7.20783895120409382744131406582, 8.650024100610536317131793058550, 9.413066739553353757875635067461, 10.69577289948008862545007257661

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