↑

Properties

Label 2-1280-80.69-c1-0-33
Degree $2$
Conductor $1280$
Sign $0.830 + 0.556i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
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  + (0.347 + 0.347i)3-s + (−0.144 − 2.23i)5-s + 3.90·7-s − 2.75i·9-s + (1.93 + 1.93i)11-s + (4.10 + 4.10i)13-s + (0.725 − 0.825i)15-s − 4.53i·17-s + (−0.0135 + 0.0135i)19-s + (1.35 + 1.35i)21-s − 6.75·23-s + (−4.95 + 0.643i)25-s + (2.00 − 2.00i)27-s + (−5.49 + 5.49i)29-s + 10.2·31-s + ⋯
L(s)  = 1  + (0.200 + 0.200i)3-s + (−0.0644 − 0.997i)5-s + 1.47·7-s − 0.919i·9-s + (0.582 + 0.582i)11-s + (1.13 + 1.13i)13-s + (0.187 − 0.213i)15-s − 1.09i·17-s + (−0.00311 + 0.00311i)19-s + (0.295 + 0.295i)21-s − 1.40·23-s + (−0.991 + 0.128i)25-s + (0.385 − 0.385i)27-s + (−1.01 + 1.01i)29-s + 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.212605693\)
\(L(\frac12)\) \(\approx\) \(2.212605693\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.144 + 2.23i)T \)
good3 \( 1 + (-0.347 - 0.347i)T + 3iT^{2} \)
7 \( 1 - 3.90T + 7T^{2} \)
11 \( 1 + (-1.93 - 1.93i)T + 11iT^{2} \)
13 \( 1 + (-4.10 - 4.10i)T + 13iT^{2} \)
17 \( 1 + 4.53iT - 17T^{2} \)
19 \( 1 + (0.0135 - 0.0135i)T - 19iT^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 + (5.49 - 5.49i)T - 29iT^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (1.72 - 1.72i)T - 37iT^{2} \)
41 \( 1 + 3.00iT - 41T^{2} \)
43 \( 1 + (-6.41 + 6.41i)T - 43iT^{2} \)
47 \( 1 + 6.57iT - 47T^{2} \)
53 \( 1 + (1.41 - 1.41i)T - 53iT^{2} \)
59 \( 1 + (-2.84 - 2.84i)T + 59iT^{2} \)
61 \( 1 + (-0.0814 + 0.0814i)T - 61iT^{2} \)
67 \( 1 + (5.90 + 5.90i)T + 67iT^{2} \)
71 \( 1 - 8.09iT - 71T^{2} \)
73 \( 1 - 0.623T + 73T^{2} \)
79 \( 1 - 4.25T + 79T^{2} \)
83 \( 1 + (-1.39 - 1.39i)T + 83iT^{2} \)
89 \( 1 + 4.17iT - 89T^{2} \)
97 \( 1 - 2.46iT - 97T^{2} \)
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

−9.292471000938093801144924325464, −8.878682206471514302685892043940, −8.185857691665324661584021242585, −7.21618699987977002385648779420, −6.24510548086596759968401773202, −5.19949577175802462352654704275, −4.35040861958534786989623930652, −3.82228663687667914153988378868, −1.99837420252587326826125144936, −1.10222362735307947530410518042, 1.41236753867006526200829721745, 2.43332199059048108902126034541, 3.62598159257979688711904450013, 4.50464356772274056644347253728, 5.83246354147666266580402351746, 6.21683311628608097474253098238, 7.72046543896799469274378867517, 7.968081660150359002547455401176, 8.567982673397096086532515103125, 9.977363627724285315037028565271

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