Properties

Label 2-1900-5.4-c1-0-8
Degree $2$
Conductor $1900$
Sign $0.894 - 0.447i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  − 1.19i·3-s + 1.19i·7-s + 1.56·9-s − 5.86·11-s + 0.364i·13-s + 1.19i·17-s + 19-s + 1.43·21-s + 8.23i·23-s − 5.46i·27-s + 7.86·29-s + 7.30·31-s + 7.03i·33-s + 7.13i·37-s + 0.436·39-s + ⋯
L(s)  = 1  − 0.692i·3-s + 0.453i·7-s + 0.521·9-s − 1.76·11-s + 0.101i·13-s + 0.290i·17-s + 0.229·19-s + 0.313·21-s + 1.71i·23-s − 1.05i·27-s + 1.46·29-s + 1.31·31-s + 1.22i·33-s + 1.17i·37-s + 0.0699·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.511586915\)
\(L(\frac12)\) \(\approx\) \(1.511586915\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 1.19iT - 3T^{2} \)
7 \( 1 - 1.19iT - 7T^{2} \)
11 \( 1 + 5.86T + 11T^{2} \)
13 \( 1 - 0.364iT - 13T^{2} \)
17 \( 1 - 1.19iT - 17T^{2} \)
23 \( 1 - 8.23iT - 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 7.30T + 31T^{2} \)
37 \( 1 - 7.13iT - 37T^{2} \)
41 \( 1 - 2.43T + 41T^{2} \)
43 \( 1 - 7.39iT - 43T^{2} \)
47 \( 1 + 13.7iT - 47T^{2} \)
53 \( 1 - 7.39iT - 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 1.30T + 61T^{2} \)
67 \( 1 - 11.9iT - 67T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 + 2.50iT - 73T^{2} \)
79 \( 1 - 7.74T + 79T^{2} \)
83 \( 1 - 3.02iT - 83T^{2} \)
89 \( 1 - 5.68T + 89T^{2} \)
97 \( 1 - 1.09iT - 97T^{2} \)
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

−9.275654801003918793214927455766, −8.147391433163440784609323255897, −7.86058123014955986971715451766, −6.97363755156023619979738768779, −6.13433610791757439883841768957, −5.28277810624601498460993332308, −4.50002116371458927227463734861, −3.13169224929395554236910406435, −2.32098954266835548243413654495, −1.13317882476894614782603636894, 0.63183222987688180841715594808, 2.37359209906776269107050269907, 3.21856652070504497290577228947, 4.52683454106616675677045127272, 4.75677383604110831092637037227, 5.87630952079067468702004219079, 6.85185378325328302268672668287, 7.68429974285222849501183845670, 8.312040730119359439292507347042, 9.274911049862706154405392385941

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