↑

Properties

Label 2-1280-40.19-c2-0-1
Degree $2$
Conductor $1280$
Sign $0.891 - 0.453i$
Analytic cond. $34.8774$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.30i·3-s + (−4.75 − 1.54i)5-s + 0.206·7-s − 19.1·9-s − 15.0·11-s − 11.6·13-s + (−8.20 + 25.2i)15-s − 18.1i·17-s − 19.3·19-s − 1.09i·21-s + 27.2·23-s + (20.2 + 14.7i)25-s + 53.6i·27-s − 44.4i·29-s + 20.3i·31-s + ⋯
L(s)  = 1  − 1.76i·3-s + (−0.950 − 0.309i)5-s + 0.0295·7-s − 2.12·9-s − 1.36·11-s − 0.899·13-s + (−0.547 + 1.68i)15-s − 1.07i·17-s − 1.02·19-s − 0.0521i·21-s + 1.18·23-s + (0.808 + 0.588i)25-s + 1.98i·27-s − 1.53i·29-s + 0.657i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ 0.891 - 0.453i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.07582080188\)
\(L(\frac12)\) \(\approx\) \(0.07582080188\)
\(L(2)\) 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 \( 1 + (4.75 + 1.54i)T \)
good3 \( 1 + 5.30iT - 9T^{2} \)
7 \( 1 - 0.206T + 49T^{2} \)
11 \( 1 + 15.0T + 121T^{2} \)
13 \( 1 + 11.6T + 169T^{2} \)
17 \( 1 + 18.1iT - 289T^{2} \)
19 \( 1 + 19.3T + 361T^{2} \)
23 \( 1 - 27.2T + 529T^{2} \)
29 \( 1 + 44.4iT - 841T^{2} \)
31 \( 1 - 20.3iT - 961T^{2} \)
37 \( 1 - 18.1T + 1.36e3T^{2} \)
41 \( 1 - 32.3T + 1.68e3T^{2} \)
43 \( 1 + 4.06iT - 1.84e3T^{2} \)
47 \( 1 - 5.37T + 2.20e3T^{2} \)
53 \( 1 - 79.1T + 2.80e3T^{2} \)
59 \( 1 + 83.3T + 3.48e3T^{2} \)
61 \( 1 - 36.7iT - 3.72e3T^{2} \)
67 \( 1 - 4.51iT - 4.48e3T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 15.5iT - 6.24e3T^{2} \)
83 \( 1 - 50.9iT - 6.88e3T^{2} \)
89 \( 1 + 10.8T + 7.92e3T^{2} \)
97 \( 1 + 12.1iT - 9.40e3T^{2} \)
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

−9.270063328742695618241321142867, −8.389193430093601420321276506800, −7.69434068203896062935270079231, −7.35067542442068270610090984030, −6.48968071780801638755191776098, −5.38724261151014847995045061788, −4.57529377480404553460913302650, −2.94393143741009669378018399596, −2.30376524404685983250621265921, −0.790508063346870422406089959621, 0.03014689718918358038164487782, 2.59731229183975550530958239957, 3.37422146522448551971967246030, 4.34691126194371572584429275957, 4.88522905638952787187368864933, 5.78220944300486672601706577295, 7.05845251513474711327043912360, 8.029926842163796025698981332824, 8.650550671427577052033560331062, 9.525664142111911694182818196619

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