Properties

Label 2-1900-19.18-c2-0-8
Degree $2$
Conductor $1900$
Sign $-0.526 + 0.850i$
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  + 5.38i·3-s + 7-s − 19.9·9-s + 14·11-s + 16.1i·13-s − 23·17-s + (10 − 16.1i)19-s + 5.38i·21-s + 23-s − 59.2i·27-s + 48.4i·29-s + 32.3i·31-s + 75.3i·33-s + 32.3i·37-s − 86.9·39-s + ⋯
L(s)  = 1  + 1.79i·3-s + 0.142·7-s − 2.22·9-s + 1.27·11-s + 1.24i·13-s − 1.35·17-s + (0.526 − 0.850i)19-s + 0.256i·21-s + 0.0434·23-s − 2.19i·27-s + 1.67i·29-s + 1.04i·31-s + 2.28i·33-s + 0.873i·37-s − 2.23·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.526 + 0.850i$
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: \((2,\ 1900,\ (\ :1),\ -0.526 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9304119833\)
\(L(\frac12)\) \(\approx\) \(0.9304119833\)
\(L(2)\) 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 \( 1 + (-10 + 16.1i)T \)
good3 \( 1 - 5.38iT - 9T^{2} \)
7 \( 1 - T + 49T^{2} \)
11 \( 1 - 14T + 121T^{2} \)
13 \( 1 - 16.1iT - 169T^{2} \)
17 \( 1 + 23T + 289T^{2} \)
23 \( 1 - T + 529T^{2} \)
29 \( 1 - 48.4iT - 841T^{2} \)
31 \( 1 - 32.3iT - 961T^{2} \)
37 \( 1 - 32.3iT - 1.36e3T^{2} \)
41 \( 1 + 32.3iT - 1.68e3T^{2} \)
43 \( 1 + 68T + 1.84e3T^{2} \)
47 \( 1 + 26T + 2.20e3T^{2} \)
53 \( 1 + 80.7iT - 2.80e3T^{2} \)
59 \( 1 + 16.1iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 + 16.1iT - 4.48e3T^{2} \)
71 \( 1 - 32.3iT - 5.04e3T^{2} \)
73 \( 1 - 7T + 5.32e3T^{2} \)
79 \( 1 + 96.9iT - 6.24e3T^{2} \)
83 \( 1 + 32T + 6.88e3T^{2} \)
89 \( 1 - 129. iT - 7.92e3T^{2} \)
97 \( 1 + 96.9iT - 9.40e3T^{2} \)
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

−9.437091758882549098576925397707, −8.940141667813688088281223377477, −8.522502019931019057223303193042, −6.87951244473126057049337685813, −6.49143340267224120843207149295, −5.03816048119746040433090629646, −4.75698720140338544383408796040, −3.83464977360317413857383659072, −3.14958904491427795788910584467, −1.70826750013309485902982299127, 0.23825991601652280007690044659, 1.28278985330950208206832815018, 2.14686116736693669525550679826, 3.16814915360480175547527920534, 4.35460927806022191141160858860, 5.70853333100271228607214661923, 6.22422608353853491382922903383, 6.91654640209174289938154260310, 7.80008402404225782701676376365, 8.185332624376198663458020955672

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