Properties

Label 2-38e2-19.18-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.229 + 0.973i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·3-s + 5-s + 7-s − 1.00·9-s − 11-s − 1.41i·15-s + 17-s − 1.41i·21-s + 1.41i·29-s + 1.41i·33-s + 35-s − 1.41i·37-s + 1.41i·41-s − 43-s − 1.00·45-s + ⋯
L(s)  = 1  − 1.41i·3-s + 5-s + 7-s − 1.00·9-s − 11-s − 1.41i·15-s + 17-s − 1.41i·21-s + 1.41i·29-s + 1.41i·33-s + 35-s − 1.41i·37-s + 1.41i·41-s − 43-s − 1.00·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375162888\)
\(L(\frac12)\) \(\approx\) \(1.375162888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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

−9.541781335649456715839484153441, −8.478877344675692620429186750915, −7.84055444481413429844057499486, −7.26487135219802101101900613453, −6.27654646058835998443996530083, −5.54853898334982717113260319787, −4.82467906905351532808468452381, −3.10830175164390682458008859562, −2.03785135530993626435906984304, −1.36050268332993543305465509124, 1.80070858114653563976834752223, 2.96572444043556335435036459743, 4.05794158105469510604608631770, 5.12110165769165580279851583405, 5.31016747309092775443413896561, 6.39945860036386387640003245147, 7.78092285616281715056948987331, 8.327991113120855842680468721897, 9.383922560218654592964385778835, 9.982688477019019676449701848876

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