Properties

Label 4-9072e2-1.1-c1e2-0-1
Degree $4$
Conductor $82301184$
Sign $1$
Analytic cond. $5247.59$
Root an. cond. $8.51118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s + 4·11-s − 4·17-s − 10·19-s − 2·23-s − 25-s + 4·29-s − 12·31-s − 4·35-s + 4·37-s − 4·43-s + 3·49-s + 12·53-s − 8·55-s − 4·59-s + 18·61-s + 16·67-s + 10·71-s − 4·73-s + 8·77-s − 6·79-s + 4·83-s + 8·85-s + 24·89-s + 20·95-s − 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 1.20·11-s − 0.970·17-s − 2.29·19-s − 0.417·23-s − 1/5·25-s + 0.742·29-s − 2.15·31-s − 0.676·35-s + 0.657·37-s − 0.609·43-s + 3/7·49-s + 1.64·53-s − 1.07·55-s − 0.520·59-s + 2.30·61-s + 1.95·67-s + 1.18·71-s − 0.468·73-s + 0.911·77-s − 0.675·79-s + 0.439·83-s + 0.867·85-s + 2.54·89-s + 2.05·95-s − 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82301184\)    =    \(2^{8} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(5247.59\)
Root analytic conductor: \(8.51118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82301184,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.700310443\)
\(L(\frac12)\) \(\approx\) \(1.700310443\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 143 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.83955062716172561656302731251, −7.82038464544903319263206966495, −6.99785508643149359926226109569, −6.87081939073847648021393907801, −6.70561195294007376400758568550, −6.33257667762718718182413818760, −5.80495920410502349463955589595, −5.55323535121953955274805441100, −5.00412268718490120571879899049, −4.79026405591668551292932033806, −4.12736257027854662369664200828, −4.11833824004431268904093119191, −3.74904118688709913280616672532, −3.61614667015159819703147009218, −2.60843945446365400167465838917, −2.45550484320101668634316809966, −1.80511105931630945182843092067, −1.75874268382395736534777293173, −0.847117957674979075798855078951, −0.35938463587698230904793883099, 0.35938463587698230904793883099, 0.847117957674979075798855078951, 1.75874268382395736534777293173, 1.80511105931630945182843092067, 2.45550484320101668634316809966, 2.60843945446365400167465838917, 3.61614667015159819703147009218, 3.74904118688709913280616672532, 4.11833824004431268904093119191, 4.12736257027854662369664200828, 4.79026405591668551292932033806, 5.00412268718490120571879899049, 5.55323535121953955274805441100, 5.80495920410502349463955589595, 6.33257667762718718182413818760, 6.70561195294007376400758568550, 6.87081939073847648021393907801, 6.99785508643149359926226109569, 7.82038464544903319263206966495, 7.83955062716172561656302731251

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