Properties

Label 2-1280-40.19-c2-0-48
Degree $2$
Conductor $1280$
Sign $-0.453 - 0.891i$
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

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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

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

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.453 - 0.891i$
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: \((2,\ 1280,\ (\ :1),\ -0.453 - 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.784822644\)
\(L(\frac12)\) \(\approx\) \(2.784822644\)
\(L(2)\) 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 \( 1 + 12.1iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.542740847806577033637837399613, −9.248700616148796400183847498071, −8.646714400457197641963638471926, −7.09991646476089340485285824574, −6.27531484489192684302690631906, −5.29591697348795556396817139048, −4.76896571616131945306549689470, −3.52055180228065528383175194747, −3.04750364491440946851549622484, −1.30715032667721752009115615736, 0.991624008725585111627323306909, 1.49629490201635295821409442912, 2.56773687018452736434320750229, 3.85752617732724473786958937867, 5.36282345337320330677257022344, 6.24380094264553046270342774041, 6.49951424375470198284580011754, 7.52423313196668228368857756641, 8.376062795084311574347940543107, 9.023190472065116459287417069010

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