↑

Properties

Label 2-1900-19.18-c2-0-27
Degree $2$
Conductor $1900$
Sign $0.943 + 0.331i$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
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  + 5.26i·3-s − 12.1·7-s − 18.7·9-s − 4.59·11-s + 19.9i·13-s + 4.10·17-s + (−17.9 − 6.30i)19-s − 64.0i·21-s − 35.2·23-s − 51.1i·27-s + 16.4i·29-s + 4.01i·31-s − 24.2i·33-s + 10.6i·37-s − 104.·39-s + ⋯
L(s)  = 1  + 1.75i·3-s − 1.73·7-s − 2.08·9-s − 0.417·11-s + 1.53i·13-s + 0.241·17-s + (−0.943 − 0.331i)19-s − 3.05i·21-s − 1.53·23-s − 1.89i·27-s + 0.568i·29-s + 0.129i·31-s − 0.733i·33-s + 0.286i·37-s − 2.68·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05361860140\)
\(L(\frac12)\) \(\approx\) \(0.05361860140\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + (17.9 + 6.30i)T \)
good3 \( 1 - 5.26iT - 9T^{2} \)
7 \( 1 + 12.1T + 49T^{2} \)
11 \( 1 + 4.59T + 121T^{2} \)
13 \( 1 - 19.9iT - 169T^{2} \)
17 \( 1 - 4.10T + 289T^{2} \)
23 \( 1 + 35.2T + 529T^{2} \)
29 \( 1 - 16.4iT - 841T^{2} \)
31 \( 1 - 4.01iT - 961T^{2} \)
37 \( 1 - 10.6iT - 1.36e3T^{2} \)
41 \( 1 + 22.4iT - 1.68e3T^{2} \)
43 \( 1 - 51.8T + 1.84e3T^{2} \)
47 \( 1 - 23.7T + 2.20e3T^{2} \)
53 \( 1 - 97.3iT - 2.80e3T^{2} \)
59 \( 1 - 33.1iT - 3.48e3T^{2} \)
61 \( 1 + 75.1T + 3.72e3T^{2} \)
67 \( 1 - 6.31iT - 4.48e3T^{2} \)
71 \( 1 + 118. iT - 5.04e3T^{2} \)
73 \( 1 + 7.02T + 5.32e3T^{2} \)
79 \( 1 + 76.0iT - 6.24e3T^{2} \)
83 \( 1 - 100.T + 6.88e3T^{2} \)
89 \( 1 - 123. iT - 7.92e3T^{2} \)
97 \( 1 - 101. iT - 9.40e3T^{2} \)
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

−9.269655946832868177034598196508, −8.655873728654722233076772481329, −7.37371056968578920642992469029, −6.30256937374523856307640532223, −5.86303763753716566955588391293, −4.62582485506676677867712272719, −4.05780908774917868827450494676, −3.31123787909862032074067293223, −2.35845521070638238622556432780, −0.02033631300878222987038385146, 0.67729280399032377703245470055, 2.16282810526751209802609488366, 2.87672333822973428975480608707, 3.80244430377944278698746077001, 5.53270759031456154241157093274, 6.12087161006746526545111043536, 6.58096026851629032051173191770, 7.62095415811997067261558468630, 7.969510814655694196952663454626, 8.869328498183357447771006067383

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