↑

Properties

Label 2-1280-40.19-c2-0-12
Degree $2$
Conductor $1280$
Sign $-0.141 - 0.989i$
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  + (−4 − 3i)5-s + 9·9-s − 24·13-s − 16i·17-s + (7 + 24i)25-s + 42i·29-s + 24·37-s + 18·41-s + (−36 − 27i)45-s − 49·49-s − 56·53-s + 22i·61-s + (96 + 72i)65-s + 96i·73-s + 81·81-s + ⋯
L(s)  = 1  + (−0.800 − 0.600i)5-s + 9-s − 1.84·13-s − 0.941i·17-s + (0.280 + 0.959i)25-s + 1.44i·29-s + 0.648·37-s + 0.439·41-s + (−0.800 − 0.599i)45-s − 0.999·49-s − 1.05·53-s + 0.360i·61-s + (1.47 + 1.10i)65-s + 1.31i·73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.141 - 0.989i$
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: \((2,\ 1280,\ (\ :1),\ -0.141 - 0.989i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7010846734\)
\(L(\frac12)\) \(\approx\) \(0.7010846734\)
\(L(2)\) 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 \( 1 + (4 + 3i)T \)
good3 \( 1 - 9T^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + 24T + 169T^{2} \)
17 \( 1 + 16iT - 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 42iT - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 24T + 1.36e3T^{2} \)
41 \( 1 - 18T + 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 56T + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 22iT - 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 96iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 - 78T + 7.92e3T^{2} \)
97 \( 1 - 144iT - 9.40e3T^{2} \)
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

−9.584539920839557319699422119091, −9.048928183708732827323776216092, −7.82971637719007988572558348077, −7.41863929300366455259484011790, −6.63781107990241485175812735820, −5.05697853509643276692118730466, −4.79821741029783324008736263543, −3.70538854698344814360187314448, −2.51348918232917226017457985967, −1.10379369077931949583608804204, 0.22622255077772809744827985490, 1.94356092839661829544362303932, 3.02867791339672783882452918539, 4.17818707283162446341061340425, 4.72381054953396964295861719865, 6.07240497627949880392792874209, 6.94352402750425767189654553222, 7.62685543525819507605177862884, 8.162893297360261222909411995941, 9.520586790995276151696775267170

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