Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s + 2·11-s − 2·19-s + 2·23-s − 5·25-s + 2·31-s + 4·35-s + 6·37-s − 10·41-s − 4·43-s + 12·47-s + 3·49-s + 8·53-s − 4·55-s + 4·59-s − 8·67-s + 14·71-s − 12·73-s − 4·77-s + 4·79-s + 16·83-s + 6·89-s + 4·95-s − 8·97-s + 8·101-s + 2·103-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 0.603·11-s − 0.458·19-s + 0.417·23-s − 25-s + 0.359·31-s + 0.676·35-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.75·47-s + 3/7·49-s + 1.09·53-s − 0.539·55-s + 0.520·59-s − 0.977·67-s + 1.66·71-s − 1.40·73-s − 0.455·77-s + 0.450·79-s + 1.75·83-s + 0.635·89-s + 0.410·95-s − 0.812·97-s + 0.796·101-s + 0.197·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.129039439$
$L(\frac12)$  $\approx$  $2.129039439$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 14 T + 189 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 178 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.088178303615233497020290640247, −7.972374870848945477812133748384, −7.42413902747773757576288558564, −7.35833796135743173611486685856, −6.68388934588475378248936281540, −6.63633932199553747540173608425, −6.17452252461083247387039738026, −5.85121525829411466575567511134, −5.34714313337098246848972402821, −5.08333890997777209408527068602, −4.40745701191563542361930427024, −4.25715347616796096079096892586, −3.74020885916209061605834357685, −3.63447077493082367195152737230, −2.97376736223391030365707243738, −2.74757531286804785731822096965, −1.94207047609305817463228247641, −1.78577032724612582119414193948, −0.66498104809541596573899391850, −0.59330392068188946780710114287, 0.59330392068188946780710114287, 0.66498104809541596573899391850, 1.78577032724612582119414193948, 1.94207047609305817463228247641, 2.74757531286804785731822096965, 2.97376736223391030365707243738, 3.63447077493082367195152737230, 3.74020885916209061605834357685, 4.25715347616796096079096892586, 4.40745701191563542361930427024, 5.08333890997777209408527068602, 5.34714313337098246848972402821, 5.85121525829411466575567511134, 6.17452252461083247387039738026, 6.63633932199553747540173608425, 6.68388934588475378248936281540, 7.35833796135743173611486685856, 7.42413902747773757576288558564, 7.972374870848945477812133748384, 8.088178303615233497020290640247

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