↑

Properties

Label 2-84e2-21.20-c1-0-73
Degree $2$
Conductor $7056$
Sign $-0.716 + 0.698i$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  + 0.765·5-s − 2i·11-s − 0.317i·13-s + 5.54·17-s − 3.69i·19-s − 3.17i·23-s − 4.41·25-s − 6.82i·29-s − 6.75i·31-s − 0.242·37-s − 2.74·41-s − 6.82·43-s − 11.9·47-s + 12.2i·53-s − 1.53i·55-s + ⋯
L(s)  = 1  + 0.342·5-s − 0.603i·11-s − 0.0879i·13-s + 1.34·17-s − 0.847i·19-s − 0.661i·23-s − 0.882·25-s − 1.26i·29-s − 1.21i·31-s − 0.0398·37-s − 0.428·41-s − 1.04·43-s − 1.74·47-s + 1.68i·53-s − 0.206i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.716 + 0.698i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.716 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.189328930\)
\(L(\frac12)\) \(\approx\) \(1.189328930\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.765T + 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 0.317iT - 13T^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 + 3.69iT - 19T^{2} \)
23 \( 1 + 3.17iT - 23T^{2} \)
29 \( 1 + 6.82iT - 29T^{2} \)
31 \( 1 + 6.75iT - 31T^{2} \)
37 \( 1 + 0.242T + 37T^{2} \)
41 \( 1 + 2.74T + 41T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 3.56iT - 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 - 9.31iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 4.32T + 83T^{2} \)
89 \( 1 - 1.66T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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

−7.83785887767848908754824794429, −6.96046522547831112018998695474, −6.12240815547893627218321815677, −5.72780066798710106439255592678, −4.85852059309605092513695860827, −4.06244911047624232706844977221, −3.18732767596628213091442436860, −2.45863979621409175291418048638, −1.39160303264584638435960655544, −0.27825117737159696435732776271, 1.40522048701305635593666524664, 1.91672715412326716306449515307, 3.34290942593788741511700983529, 3.55017919639224826877897011787, 4.99273549959022959125777386710, 5.14782316782371033432832338635, 6.20159725570257862110846841083, 6.71514482818105549512161521432, 7.67306393455864437337147836433, 7.999621359795691522207314784457

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