↑

Properties

Label 2-1344-48.11-c1-0-27
Degree $2$
Conductor $1344$
Sign $0.683 + 0.730i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
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  + (−1.72 + 0.119i)3-s + (0.793 − 0.793i)5-s + 7-s + (2.97 − 0.414i)9-s + (0.687 + 0.687i)11-s + (2.91 − 2.91i)13-s + (−1.27 + 1.46i)15-s − 1.81i·17-s + (−3.24 − 3.24i)19-s + (−1.72 + 0.119i)21-s + 5.92i·23-s + 3.74i·25-s + (−5.08 + 1.07i)27-s + (1.54 + 1.54i)29-s − 3.60i·31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0691i)3-s + (0.354 − 0.354i)5-s + 0.377·7-s + (0.990 − 0.138i)9-s + (0.207 + 0.207i)11-s + (0.807 − 0.807i)13-s + (−0.329 + 0.378i)15-s − 0.441i·17-s + (−0.743 − 0.743i)19-s + (−0.377 + 0.0261i)21-s + 1.23i·23-s + 0.748i·25-s + (−0.978 + 0.206i)27-s + (0.287 + 0.287i)29-s − 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.683 + 0.730i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.683 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.328763454\)
\(L(\frac12)\) \(\approx\) \(1.328763454\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.72 - 0.119i)T \)
7 \( 1 - T \)
good5 \( 1 + (-0.793 + 0.793i)T - 5iT^{2} \)
11 \( 1 + (-0.687 - 0.687i)T + 11iT^{2} \)
13 \( 1 + (-2.91 + 2.91i)T - 13iT^{2} \)
17 \( 1 + 1.81iT - 17T^{2} \)
19 \( 1 + (3.24 + 3.24i)T + 19iT^{2} \)
23 \( 1 - 5.92iT - 23T^{2} \)
29 \( 1 + (-1.54 - 1.54i)T + 29iT^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (5.10 + 5.10i)T + 37iT^{2} \)
41 \( 1 - 1.19T + 41T^{2} \)
43 \( 1 + (-7.00 + 7.00i)T - 43iT^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 + (2.08 - 2.08i)T - 53iT^{2} \)
59 \( 1 + (-0.919 - 0.919i)T + 59iT^{2} \)
61 \( 1 + (-4.21 + 4.21i)T - 61iT^{2} \)
67 \( 1 + (7.87 + 7.87i)T + 67iT^{2} \)
71 \( 1 + 7.99iT - 71T^{2} \)
73 \( 1 - 4.19iT - 73T^{2} \)
79 \( 1 + 8.07iT - 79T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 - 7.73T + 89T^{2} \)
97 \( 1 - 6.61T + 97T^{2} \)
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

−9.414384264093844478974200093867, −8.945145279603482120000614595983, −7.70620367376282251235730242512, −7.04959858435823791111332808170, −5.94206478687668764235912093098, −5.45475494525916029114917590199, −4.56529322176317775571043841021, −3.56836695179254391698578436919, −1.95186961939397039641637481381, −0.75047940865321845875652484403, 1.16846172180491040783503969005, 2.32254100188119096065504200277, 3.94261963875331568908029776885, 4.58570415585630857494223575041, 5.79659311658891900813285015355, 6.33243070567936293635648255277, 6.97557922183759863457690318448, 8.165071066278614505050927144833, 8.843372833501213672652208680481, 9.989983406507232531848633115054

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