Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \cdot 7^{2} $
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  − 6·5-s − 7-s − 6·11-s − 5·19-s + 12·23-s + 19·25-s + 5·31-s + 6·35-s + 11·37-s − 6·47-s − 6·49-s + 12·53-s + 36·55-s + 12·59-s − 24·61-s − 15·67-s − 9·73-s + 6·77-s − 21·79-s − 36·83-s − 12·89-s + 30·95-s − 18·101-s + 5·103-s + 7·109-s + 12·113-s − 72·115-s + ⋯
L(s)  = 1  − 2.68·5-s − 0.377·7-s − 1.80·11-s − 1.14·19-s + 2.50·23-s + 19/5·25-s + 0.898·31-s + 1.01·35-s + 1.80·37-s − 0.875·47-s − 6/7·49-s + 1.64·53-s + 4.85·55-s + 1.56·59-s − 3.07·61-s − 1.83·67-s − 1.05·73-s + 0.683·77-s − 2.36·79-s − 3.95·83-s − 1.27·89-s + 3.07·95-s − 1.79·101-s + 0.492·103-s + 0.670·109-s + 1.12·113-s − 6.71·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2331204502\)
\(L(\frac12)\)  \(\approx\)  \(0.2331204502\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−10.28287541406076884592324924230, −9.844461022451727615447584396806, −9.270469766163606916296520725264, −8.565486820617685523328977339289, −8.425441559292425038998015201861, −8.274319603405284888248795870631, −7.53251725893981925389476069566, −7.24211190464527637064942741586, −7.22015101156579925092374443938, −6.48619652650809141664367753604, −5.83900768137407154594884121638, −5.38534997325518752022333756957, −4.59179730223089507811351871447, −4.43074147001037008747509306031, −4.20207987542667339646694479725, −3.12930014457128871505268877591, −3.05764118467688180907451683891, −2.65175136502634313387120931747, −1.24864156146184720534655393394, −0.24207554647875397757989957833, 0.24207554647875397757989957833, 1.24864156146184720534655393394, 2.65175136502634313387120931747, 3.05764118467688180907451683891, 3.12930014457128871505268877591, 4.20207987542667339646694479725, 4.43074147001037008747509306031, 4.59179730223089507811351871447, 5.38534997325518752022333756957, 5.83900768137407154594884121638, 6.48619652650809141664367753604, 7.22015101156579925092374443938, 7.24211190464527637064942741586, 7.53251725893981925389476069566, 8.274319603405284888248795870631, 8.425441559292425038998015201861, 8.565486820617685523328977339289, 9.270469766163606916296520725264, 9.844461022451727615447584396806, 10.28287541406076884592324924230

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