Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·11-s − 4·13-s − 4·19-s − 6·23-s + 5·25-s − 12·29-s + 8·31-s − 2·37-s + 24·41-s − 8·43-s − 12·47-s − 6·53-s − 10·61-s − 8·67-s − 12·71-s − 10·73-s + 4·79-s − 24·83-s − 12·89-s + 20·97-s + 12·101-s + 8·103-s − 6·107-s − 14·109-s + 12·113-s + 11·121-s + 127-s + ⋯
L(s)  = 1  − 1.80·11-s − 1.10·13-s − 0.917·19-s − 1.25·23-s + 25-s − 2.22·29-s + 1.43·31-s − 0.328·37-s + 3.74·41-s − 1.21·43-s − 1.75·47-s − 0.824·53-s − 1.28·61-s − 0.977·67-s − 1.42·71-s − 1.17·73-s + 0.450·79-s − 2.63·83-s − 1.27·89-s + 2.03·97-s + 1.19·101-s + 0.788·103-s − 0.580·107-s − 1.34·109-s + 1.12·113-s + 121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2284518589\)
\(L(\frac12)\)  \(\approx\)  \(0.2284518589\)
\(L(\frac{3}{2})\)   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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.554574195003549911579618090288, −9.057669002120242481862504988676, −8.773650660616631093854656284318, −8.115740235985354181582674125923, −7.960665559929633233705723364521, −7.43459518678917806391853825802, −7.42644860065159425037676445151, −6.72575848984387405197821086068, −6.20080554217265571732577823089, −5.74946873491688202489108732054, −5.64655303408395581646025433378, −4.76355785055075911897485270082, −4.69279063319515618283618631482, −4.27818700991027018142613131213, −3.55129892550125268171494229195, −2.85389113381592196300907702039, −2.68751287498481909050364269498, −2.07900445752771484411713967524, −1.46082212216262558740401482019, −0.17057248429115436909393366527, 0.17057248429115436909393366527, 1.46082212216262558740401482019, 2.07900445752771484411713967524, 2.68751287498481909050364269498, 2.85389113381592196300907702039, 3.55129892550125268171494229195, 4.27818700991027018142613131213, 4.69279063319515618283618631482, 4.76355785055075911897485270082, 5.64655303408395581646025433378, 5.74946873491688202489108732054, 6.20080554217265571732577823089, 6.72575848984387405197821086068, 7.42644860065159425037676445151, 7.43459518678917806391853825802, 7.960665559929633233705723364521, 8.115740235985354181582674125923, 8.773650660616631093854656284318, 9.057669002120242481862504988676, 9.554574195003549911579618090288

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