↑

Properties

Label 2-1200-48.5-c0-0-0
Degree $2$
Conductor $1200$
Sign $0.382 + 0.923i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)12-s − 1.00·16-s + 1.41i·17-s + (0.707 − 0.707i)18-s + (1 − i)19-s + 1.41·23-s − 1.00·24-s + (0.707 − 0.707i)27-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)12-s − 1.00·16-s + 1.41i·17-s + (0.707 − 0.707i)18-s + (1 − i)19-s + 1.41·23-s − 1.00·24-s + (0.707 − 0.707i)27-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5999395315\)
\(L(\frac12)\) \(\approx\) \(0.5999395315\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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

−9.963946576342807016962725960992, −8.936598007806978755845485294165, −8.266593806846385204500880192796, −7.29559396696789634265742088854, −6.78944905782335106903409069493, −5.61352216969456553572159878063, −4.61261543005880542674168139020, −3.35123632032944131225093329490, −2.18684375825663080811168665870, −1.01483283962229245839022808768, 1.06212636651580308915662667725, 2.97651785823872670969237959653, 4.37948971945085264098297565671, 5.25136674907686538888713125825, 5.82519521958052877072932764571, 6.90586631075897811206785935729, 7.47408814092031433367239950892, 8.637355040272343701305236274743, 9.391011093691416634401205100560, 9.891352024586910173243018523983

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