Properties

Degree 8
Conductor $ 2^{16} \cdot 3^{12} \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  − 4·7-s − 14·25-s + 20·37-s − 20·43-s − 2·49-s − 8·67-s + 52·79-s − 28·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 56·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 1.51·7-s − 2.79·25-s + 3.28·37-s − 3.04·43-s − 2/7·49-s − 0.977·67-s + 5.85·79-s − 2.68·109-s + 3.63·121-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 + 3.07·169-s + 0.0760·173-s + 4.23·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.610633978\)
\(L(\frac12)\)  \(\approx\)  \(1.610633978\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.21397935896695002107825542070, −6.13996899913554701368300562726, −5.73531807293022851674778658168, −5.71489816820015842887008912033, −5.37230710686915281675145381744, −5.06758980826642576927217905432, −5.04955768075865872459875128840, −4.86604173409420923556668248055, −4.39397653356963753801259640463, −4.24918047807647133560683929828, −4.16310535481772953083875700005, −3.85211206974430823711303568972, −3.55708490797448211037227225228, −3.48785239393292837442239817350, −3.19701259060450541425572749375, −3.11923114137449066387456638536, −2.64485509824527501329535462750, −2.53391645658292052249275060187, −2.22086211762820082792330300084, −1.95575135540242145309739374218, −1.71976668654173784339913194175, −1.42506369053523215070259667415, −0.946872340514202530897716208534, −0.50176122247856253311475649149, −0.29673155227346828015419320557, 0.29673155227346828015419320557, 0.50176122247856253311475649149, 0.946872340514202530897716208534, 1.42506369053523215070259667415, 1.71976668654173784339913194175, 1.95575135540242145309739374218, 2.22086211762820082792330300084, 2.53391645658292052249275060187, 2.64485509824527501329535462750, 3.11923114137449066387456638536, 3.19701259060450541425572749375, 3.48785239393292837442239817350, 3.55708490797448211037227225228, 3.85211206974430823711303568972, 4.16310535481772953083875700005, 4.24918047807647133560683929828, 4.39397653356963753801259640463, 4.86604173409420923556668248055, 5.04955768075865872459875128840, 5.06758980826642576927217905432, 5.37230710686915281675145381744, 5.71489816820015842887008912033, 5.73531807293022851674778658168, 6.13996899913554701368300562726, 6.21397935896695002107825542070

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