Properties

Label 2-38e2-4.3-c0-0-1
Degree $2$
Conductor $1444$
Sign $1$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.618·5-s − 8-s + 9-s − 0.618·10-s + 1.61·13-s + 16-s − 1.61·17-s − 18-s + 0.618·20-s − 0.618·25-s − 1.61·26-s + 1.61·29-s − 32-s + 1.61·34-s + 36-s − 0.618·37-s − 0.618·40-s − 0.618·41-s + 0.618·45-s + 49-s + 0.618·50-s + 1.61·52-s − 0.618·53-s − 1.61·58-s − 1.61·61-s + ⋯
L(s)  = 1  − 2-s + 4-s + 0.618·5-s − 8-s + 9-s − 0.618·10-s + 1.61·13-s + 16-s − 1.61·17-s − 18-s + 0.618·20-s − 0.618·25-s − 1.61·26-s + 1.61·29-s − 32-s + 1.61·34-s + 36-s − 0.618·37-s − 0.618·40-s − 0.618·41-s + 0.618·45-s + 49-s + 0.618·50-s + 1.61·52-s − 0.618·53-s − 1.61·58-s − 1.61·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (723, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8828567398\)
\(L(\frac12)\) \(\approx\) \(0.8828567398\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 \)
good3 \( 1 - T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.61T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 + 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.618T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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

−9.689242347296838994365521450259, −8.878353664731854470088932679646, −8.383454299978431655208036264200, −7.31510960255196794872843257777, −6.49680650443331888950116910231, −6.04675445483723202641328081025, −4.67114994899042061909186054176, −3.55170664133589650237045826662, −2.23037934588408357862289591024, −1.31287046705749264617439250942, 1.31287046705749264617439250942, 2.23037934588408357862289591024, 3.55170664133589650237045826662, 4.67114994899042061909186054176, 6.04675445483723202641328081025, 6.49680650443331888950116910231, 7.31510960255196794872843257777, 8.383454299978431655208036264200, 8.878353664731854470088932679646, 9.689242347296838994365521450259

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