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  + 3·5-s + 7-s − 3·9-s − 6·11-s − 2·13-s + 12·17-s + 14·19-s + 3·23-s + 5·25-s − 6·29-s + 2·31-s + 3·35-s + 4·37-s + 2·43-s − 9·45-s + 12·53-s − 18·55-s − 5·61-s − 3·63-s − 6·65-s + 8·67-s − 6·71-s + 4·73-s − 6·77-s + 5·79-s + 9·81-s + 12·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 9-s − 1.80·11-s − 0.554·13-s + 2.91·17-s + 3.21·19-s + 0.625·23-s + 25-s − 1.11·29-s + 0.359·31-s + 0.507·35-s + 0.657·37-s + 0.304·43-s − 1.34·45-s + 1.64·53-s − 2.42·55-s − 0.640·61-s − 0.377·63-s − 0.744·65-s + 0.977·67-s − 0.712·71-s + 0.468·73-s − 0.683·77-s + 0.562·79-s + 81-s + 1.31·83-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  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.945529364\)
\(L(\frac12)\)  \(\approx\)  \(2.945529364\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−10.10988741422847538334413575735, −9.673652374575824079255213528516, −9.383947140078458358520778779524, −9.307061164784269253296220964657, −8.377156827410472067792029340427, −7.903983053165542946697583603814, −7.79846340776504668548893833396, −7.36426437142942125919744457445, −6.97158293428436286731633916590, −6.01251393840904466698967811897, −5.69234507888719946771370170362, −5.43571162268061002545060068987, −5.15389003042644954330046899002, −4.94992987742234321206390062877, −3.63935879178098036372027016555, −3.30663373907994560975155351255, −2.63813215317811622380824500758, −2.55325418531545072069682681972, −1.39981180678212039421183646151, −0.872297971572014747466251686579, 0.872297971572014747466251686579, 1.39981180678212039421183646151, 2.55325418531545072069682681972, 2.63813215317811622380824500758, 3.30663373907994560975155351255, 3.63935879178098036372027016555, 4.94992987742234321206390062877, 5.15389003042644954330046899002, 5.43571162268061002545060068987, 5.69234507888719946771370170362, 6.01251393840904466698967811897, 6.97158293428436286731633916590, 7.36426437142942125919744457445, 7.79846340776504668548893833396, 7.903983053165542946697583603814, 8.377156827410472067792029340427, 9.307061164784269253296220964657, 9.383947140078458358520778779524, 9.673652374575824079255213528516, 10.10988741422847538334413575735

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