Properties

Label 2-84e2-28.27-c1-0-95
Degree $2$
Conductor $7056$
Sign $-0.777 - 0.629i$
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  − 4.29i·5-s − 5.06i·11-s + 3.37i·13-s − 2.76i·17-s + 4.70·19-s + 4.83i·23-s − 13.4·25-s − 2.46·29-s − 5.69·31-s − 2.33·37-s + 5.14i·41-s − 13.0i·43-s − 5.35·47-s + 4.22·53-s − 21.7·55-s + ⋯
L(s)  = 1  − 1.92i·5-s − 1.52i·11-s + 0.937i·13-s − 0.670i·17-s + 1.07·19-s + 1.00i·23-s − 2.69·25-s − 0.456·29-s − 1.02·31-s − 0.383·37-s + 0.804i·41-s − 1.98i·43-s − 0.781·47-s + 0.580·53-s − 2.93·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.777 - 0.629i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.777 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7624685654\)
\(L(\frac12)\) \(\approx\) \(0.7624685654\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.29iT - 5T^{2} \)
11 \( 1 + 5.06iT - 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 + 2.76iT - 17T^{2} \)
19 \( 1 - 4.70T + 19T^{2} \)
23 \( 1 - 4.83iT - 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 5.69T + 31T^{2} \)
37 \( 1 + 2.33T + 37T^{2} \)
41 \( 1 - 5.14iT - 41T^{2} \)
43 \( 1 + 13.0iT - 43T^{2} \)
47 \( 1 + 5.35T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 + 9.60T + 59T^{2} \)
61 \( 1 + 3.87iT - 61T^{2} \)
67 \( 1 + 4.76iT - 67T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 - 13.9iT - 79T^{2} \)
83 \( 1 + 7.32T + 83T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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

−7.66936717056637275882414776655, −6.88777689112456729375236112690, −5.73394857689199771225436252621, −5.47715801217575704838406492371, −4.77946519330592672927957720823, −3.89898551221260448311674552325, −3.27253584553188334345902422322, −1.87496522483734251925045749437, −1.13434277924406988275178192561, −0.18805878077539160651344737733, 1.64595592179805648204813733988, 2.49836074672839114483913709126, 3.16242017882259825489799057814, 3.87109659238748465688559305371, 4.80028701182343437893479251775, 5.73807903464256025156805364060, 6.34901288450616913301460119567, 7.10640829746016872046657667681, 7.48217770810834097619799172357, 8.052676169113394937505181348061

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