↑

Properties

Label 2-1200-5.2-c2-0-8
Degree $2$
Conductor $1200$
Sign $-0.229 - 0.973i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s + (7.33 + 7.33i)7-s − 2.99i·9-s − 6.49·11-s + (−11.2 + 11.2i)13-s + (10.1 + 10.1i)17-s − 6.67i·19-s + 17.9·21-s + (−21.0 + 21.0i)23-s + (−3.67 − 3.67i)27-s + 12.2i·29-s − 46.1·31-s + (−7.95 + 7.95i)33-s + (48.6 + 48.6i)37-s + 27.5i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.04 + 1.04i)7-s − 0.333i·9-s − 0.590·11-s + (−0.864 + 0.864i)13-s + (0.596 + 0.596i)17-s − 0.351i·19-s + 0.856·21-s + (−0.916 + 0.916i)23-s + (−0.136 − 0.136i)27-s + 0.421i·29-s − 1.49·31-s + (−0.241 + 0.241i)33-s + (1.31 + 1.31i)37-s + 0.705i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.579992219\)
\(L(\frac12)\) \(\approx\) \(1.579992219\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-7.33 - 7.33i)T + 49iT^{2} \)
11 \( 1 + 6.49T + 121T^{2} \)
13 \( 1 + (11.2 - 11.2i)T - 169iT^{2} \)
17 \( 1 + (-10.1 - 10.1i)T + 289iT^{2} \)
19 \( 1 + 6.67iT - 361T^{2} \)
23 \( 1 + (21.0 - 21.0i)T - 529iT^{2} \)
29 \( 1 - 12.2iT - 841T^{2} \)
31 \( 1 + 46.1T + 961T^{2} \)
37 \( 1 + (-48.6 - 48.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 60.0T + 1.68e3T^{2} \)
43 \( 1 + (-22.1 + 22.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (51.3 + 51.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (42.4 - 42.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 0.297iT - 3.48e3T^{2} \)
61 \( 1 - 46.8T + 3.72e3T^{2} \)
67 \( 1 + (13.4 + 13.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 46.3T + 5.04e3T^{2} \)
73 \( 1 + (-42.4 + 42.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 27.7iT - 6.24e3T^{2} \)
83 \( 1 + (-71.7 + 71.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 69.5iT - 7.92e3T^{2} \)
97 \( 1 + (-110. - 110. i)T + 9.40e3iT^{2} \)
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

−9.610833679726763111024246618267, −8.916547292771217585618897079229, −8.064076858946679034857323218935, −7.58349039984480550536122107370, −6.50065346775850589954812772782, −5.46578026238338524705194066967, −4.82148945937893876155617666086, −3.52323457159174052547869613625, −2.29761267549241239004906712523, −1.63989444014911121086235730723, 0.41214445570416227748778875565, 1.92801912946765862497774348006, 3.07460112527400656716140940266, 4.17301852610239787609378325545, 4.90689130944446266790117231404, 5.73768036787798973072777453975, 7.16166036852961419964910586386, 7.84333407224452070622637820595, 8.210790402401645596918895992727, 9.525025318112160814004073186280

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