Properties

Label 2-168e2-1.1-c1-0-1
Degree $2$
Conductor $28224$
Sign $1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·11-s − 6·13-s − 5·17-s − 19-s − 7·23-s − 4·25-s + 2·29-s − 5·31-s − 3·37-s − 2·41-s − 4·43-s − 5·47-s − 53-s − 3·55-s + 15·59-s − 5·61-s − 6·65-s − 9·67-s − 7·73-s − 79-s + 12·83-s − 5·85-s + 7·89-s − 95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.904·11-s − 1.66·13-s − 1.21·17-s − 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.371·29-s − 0.898·31-s − 0.493·37-s − 0.312·41-s − 0.609·43-s − 0.729·47-s − 0.137·53-s − 0.404·55-s + 1.95·59-s − 0.640·61-s − 0.744·65-s − 1.09·67-s − 0.819·73-s − 0.112·79-s + 1.31·83-s − 0.542·85-s + 0.741·89-s − 0.102·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{28224} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3682960629\)
\(L(\frac12)\) \(\approx\) \(0.3682960629\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−15.16584605855201, −14.72346437298181, −14.14893252028528, −13.55158450963127, −13.15541046441091, −12.60636904674830, −11.94352331569466, −11.62859221073525, −10.75830811630059, −10.28621510432262, −9.869060102343081, −9.337613937258824, −8.642308322331607, −8.023558467077805, −7.513478539440773, −6.899434734675473, −6.292412750617588, −5.588357157682364, −5.054618727776293, −4.485637998554770, −3.766972621333801, −2.829773799802868, −2.185047850671946, −1.806196979329178, −0.2137652860221413, 0.2137652860221413, 1.806196979329178, 2.185047850671946, 2.829773799802868, 3.766972621333801, 4.485637998554770, 5.054618727776293, 5.588357157682364, 6.292412750617588, 6.899434734675473, 7.513478539440773, 8.023558467077805, 8.642308322331607, 9.337613937258824, 9.869060102343081, 10.28621510432262, 10.75830811630059, 11.62859221073525, 11.94352331569466, 12.60636904674830, 13.15541046441091, 13.55158450963127, 14.14893252028528, 14.72346437298181, 15.16584605855201

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