Properties

Degree $4$
Conductor $4064256$
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  − 4·5-s − 5·7-s + 6·11-s + 10·13-s + 2·17-s − 19-s − 6·23-s + 5·25-s + 3·31-s + 20·35-s − 3·37-s + 12·41-s + 10·43-s − 4·47-s + 18·49-s − 6·53-s − 24·55-s − 6·59-s + 2·61-s − 40·65-s − 7·67-s − 32·71-s + 3·73-s − 30·77-s − 11·79-s − 24·83-s − 8·85-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.88·7-s + 1.80·11-s + 2.77·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 25-s + 0.538·31-s + 3.38·35-s − 0.493·37-s + 1.87·41-s + 1.52·43-s − 0.583·47-s + 18/7·49-s − 0.824·53-s − 3.23·55-s − 0.781·59-s + 0.256·61-s − 4.96·65-s − 0.855·67-s − 3.79·71-s + 0.351·73-s − 3.41·77-s − 1.23·79-s − 2.63·83-s − 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4064256\)    =    \(2^{10} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4064256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.172916437\)
\(L(\frac12)\) \(\approx\) \(1.172916437\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−9.705669078799358469279935259753, −8.874871154837142022026726549725, −8.748867424409151745968061019412, −8.228226317465367332758961560617, −7.76732161916545072025508817617, −7.32443263578210754350423006043, −7.12186113737288095602389529301, −6.44986139641714886459493366663, −6.14073094259087855165751058282, −5.91783845579342137278276328283, −5.81357087601946103823510898100, −4.40511545942400333726023419678, −4.29361359603048031755335932096, −3.93509239316057844557849880793, −3.68102080081483078686252494703, −3.09834762131747131662728026251, −2.98474324475393554791208926417, −1.71464561998985268449813331184, −1.15789215053590767730341613715, −0.45630287248820541482970382664, 0.45630287248820541482970382664, 1.15789215053590767730341613715, 1.71464561998985268449813331184, 2.98474324475393554791208926417, 3.09834762131747131662728026251, 3.68102080081483078686252494703, 3.93509239316057844557849880793, 4.29361359603048031755335932096, 4.40511545942400333726023419678, 5.81357087601946103823510898100, 5.91783845579342137278276328283, 6.14073094259087855165751058282, 6.44986139641714886459493366663, 7.12186113737288095602389529301, 7.32443263578210754350423006043, 7.76732161916545072025508817617, 8.228226317465367332758961560617, 8.748867424409151745968061019412, 8.874871154837142022026726549725, 9.705669078799358469279935259753

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