Properties

Label 2-168e2-1.1-c1-0-110
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 5·11-s − 2·13-s − 6·17-s − 2·19-s + 6·23-s − 4·25-s − 3·29-s + 5·31-s + 2·37-s − 8·41-s + 4·43-s − 4·47-s + 9·53-s − 5·55-s − 3·59-s + 12·61-s + 2·65-s − 2·67-s + 8·71-s − 14·73-s + 79-s + 17·83-s + 6·85-s − 18·89-s + 2·95-s + 3·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.557·29-s + 0.898·31-s + 0.328·37-s − 1.24·41-s + 0.609·43-s − 0.583·47-s + 1.23·53-s − 0.674·55-s − 0.390·59-s + 1.53·61-s + 0.248·65-s − 0.244·67-s + 0.949·71-s − 1.63·73-s + 0.112·79-s + 1.86·83-s + 0.650·85-s − 1.90·89-s + 0.205·95-s + 0.304·97-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: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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

−15.26812174514793, −15.01754452334082, −14.60097890797473, −13.78402629549630, −13.43089684786087, −12.82226770046420, −12.17494666708591, −11.69428572131417, −11.28773744315046, −10.78925073507896, −9.960662226965374, −9.507569926094559, −8.826588222928495, −8.562308190486280, −7.757345972844551, −6.993502795661827, −6.744612830912135, −6.111258940967210, −5.276623006612760, −4.553729697380832, −4.106336705810196, −3.503216651398006, −2.598161014998271, −1.909715473125982, −1.008526393291822, 0, 1.008526393291822, 1.909715473125982, 2.598161014998271, 3.503216651398006, 4.106336705810196, 4.553729697380832, 5.276623006612760, 6.111258940967210, 6.744612830912135, 6.993502795661827, 7.757345972844551, 8.562308190486280, 8.826588222928495, 9.507569926094559, 9.960662226965374, 10.78925073507896, 11.28773744315046, 11.69428572131417, 12.17494666708591, 12.82226770046420, 13.43089684786087, 13.78402629549630, 14.60097890797473, 15.01754452334082, 15.26812174514793

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