Properties

Degree $4$
Conductor $2458624$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 2·9-s − 4·11-s − 6·13-s − 4·15-s − 2·19-s + 8·23-s − 2·25-s + 6·27-s − 4·31-s − 8·33-s − 12·39-s + 8·41-s + 4·43-s − 4·45-s − 12·47-s − 20·53-s + 8·55-s − 4·57-s + 14·59-s − 18·61-s + 12·65-s + 8·67-s + 16·69-s + 8·71-s − 12·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 2/3·9-s − 1.20·11-s − 1.66·13-s − 1.03·15-s − 0.458·19-s + 1.66·23-s − 2/5·25-s + 1.15·27-s − 0.718·31-s − 1.39·33-s − 1.92·39-s + 1.24·41-s + 0.609·43-s − 0.596·45-s − 1.75·47-s − 2.74·53-s + 1.07·55-s − 0.529·57-s + 1.82·59-s − 2.30·61-s + 1.48·65-s + 0.977·67-s + 1.92·69-s + 0.949·71-s − 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1568} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.323772904\)
\(L(\frac12)\) \(\approx\) \(1.323772904\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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

−9.529362744689139126816148281944, −9.381330686401094257156954132096, −8.705412889654435934657014059545, −8.423572559963947702829390767235, −7.889147343207897363008561799069, −7.81510927903905389766787600569, −7.28831557806710236450435715403, −7.17800403749017692338791301821, −6.42839971483428242832740500367, −6.13871497065221366604902274811, −5.12614524957281905504214628061, −5.11076632810424960799315498835, −4.64974042908500062617369353778, −4.16898783984654253171752289648, −3.40510413960881198425224142312, −3.23584617740481828152951930413, −2.50828880986518194453449779751, −2.41762720435726300677545954570, −1.51418020789488514632355481407, −0.40506550031819905158841571449, 0.40506550031819905158841571449, 1.51418020789488514632355481407, 2.41762720435726300677545954570, 2.50828880986518194453449779751, 3.23584617740481828152951930413, 3.40510413960881198425224142312, 4.16898783984654253171752289648, 4.64974042908500062617369353778, 5.11076632810424960799315498835, 5.12614524957281905504214628061, 6.13871497065221366604902274811, 6.42839971483428242832740500367, 7.17800403749017692338791301821, 7.28831557806710236450435715403, 7.81510927903905389766787600569, 7.889147343207897363008561799069, 8.423572559963947702829390767235, 8.705412889654435934657014059545, 9.381330686401094257156954132096, 9.529362744689139126816148281944

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