Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \cdot 7^{4} $
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  − 8·11-s − 16·23-s − 8·25-s + 4·29-s + 8·37-s + 8·43-s + 8·53-s + 24·67-s + 32·79-s + 8·107-s + 8·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2.41·11-s − 3.33·23-s − 8/5·25-s + 0.742·29-s + 1.31·37-s + 1.21·43-s + 1.09·53-s + 2.93·67-s + 3.60·79-s + 0.773·107-s + 0.766·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7056} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.698102060\)
\(L(\frac12)\)  \(\approx\)  \(1.698102060\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 96 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.000986055808044704131864045510, −7.969211615481351891843541775523, −7.47886674305724041792151187021, −7.32255808568705907177971119939, −6.57810833253111255509064167535, −6.32301369786361087113965678331, −5.93918724800471733548830556486, −5.73132766061274245836021446321, −5.21644729068976999674154883021, −5.15180032184453151614871203081, −4.34696937816104427224743853008, −4.31886659857370845762107413181, −3.62569051648689026834325333295, −3.60654459229780335061141967597, −2.67698996935937688268733183432, −2.51951359637012108384803085445, −1.99231613645070061606911589619, −1.96332279105174111741697194843, −0.70026480379897881243970751324, −0.44492244615190841629398251278, 0.44492244615190841629398251278, 0.70026480379897881243970751324, 1.96332279105174111741697194843, 1.99231613645070061606911589619, 2.51951359637012108384803085445, 2.67698996935937688268733183432, 3.60654459229780335061141967597, 3.62569051648689026834325333295, 4.31886659857370845762107413181, 4.34696937816104427224743853008, 5.15180032184453151614871203081, 5.21644729068976999674154883021, 5.73132766061274245836021446321, 5.93918724800471733548830556486, 6.32301369786361087113965678331, 6.57810833253111255509064167535, 7.32255808568705907177971119939, 7.47886674305724041792151187021, 7.969211615481351891843541775523, 8.000986055808044704131864045510

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