↑

Properties

Label 2-2736-57.8-c1-0-3
Degree $2$
Conductor $2736$
Sign $0.331 - 0.943i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + (−1.76 − 1.01i)5-s − 3.69·7-s + 0.605i·11-s + (2.65 − 1.53i)13-s + (−2.40 − 1.39i)17-s + (−0.780 − 4.28i)19-s + (−2.23 + 1.29i)23-s + (−0.422 − 0.732i)25-s + (1.70 + 2.95i)29-s + 4.42i·31-s + (6.52 + 3.76i)35-s − 3.34i·37-s + (1.50 − 2.60i)41-s + (−0.562 + 0.974i)43-s + (−3.05 + 1.76i)47-s + ⋯
L(s)  = 1  + (−0.789 − 0.455i)5-s − 1.39·7-s + 0.182i·11-s + (0.735 − 0.424i)13-s + (−0.584 − 0.337i)17-s + (−0.179 − 0.983i)19-s + (−0.466 + 0.269i)23-s + (−0.0845 − 0.146i)25-s + (0.316 + 0.548i)29-s + 0.795i·31-s + (1.10 + 0.637i)35-s − 0.550i·37-s + (0.234 − 0.406i)41-s + (−0.0858 + 0.148i)43-s + (−0.445 + 0.257i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6196507796\)
\(L(\frac12)\) \(\approx\) \(0.6196507796\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (0.780 + 4.28i)T \)
good5 \( 1 + (1.76 + 1.01i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 - 0.605iT - 11T^{2} \)
13 \( 1 + (-2.65 + 1.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.40 + 1.39i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.23 - 1.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.70 - 2.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.42iT - 31T^{2} \)
37 \( 1 + 3.34iT - 37T^{2} \)
41 \( 1 + (-1.50 + 2.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.562 - 0.974i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.05 - 1.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.984 - 1.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.48 - 6.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.98 - 8.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.324 - 0.187i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.03 - 13.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.09 + 14.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.44 - 4.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.17iT - 83T^{2} \)
89 \( 1 + (-3.91 - 6.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 - 4.95i)T + (48.5 + 84.0i)T^{2} \)
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

−8.951918083235552665429812719491, −8.335187045774096686379949741965, −7.38924332490307734347075758553, −6.72266780173853471953362605025, −6.01529841696793770306959345628, −5.01627317349849563742048669197, −4.11586215348943315240144998460, −3.40653069317318417088904875306, −2.49723273412165989367633621141, −0.849781793084375534362156410359, 0.26733706383186992936791185551, 1.96537329714957331793205805152, 3.22300241309575805529489203090, 3.69902273561652768517210811835, 4.51160407085503272236214718141, 5.95470256933152068910406117525, 6.32047662233025737590683016927, 7.07149223306137156523202517828, 7.984165012181122501875939716798, 8.564014000804127564171856120219

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