Properties

Degree 8
Conductor $ 2^{28} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 132·9-s − 648·17-s + 2.37e3·25-s + 7.56e3·41-s − 2.68e3·49-s + 1.10e4·73-s − 54·81-s + 9.72e3·89-s + 2.98e4·97-s − 2.80e3·113-s − 3.53e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8.55e4·153-s + 157-s + 163-s + 167-s + 2.09e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1.62·9-s − 2.24·17-s + 3.79·25-s + 4.49·41-s − 1.11·49-s + 2.06·73-s − 0.00823·81-s + 1.22·89-s + 3.16·97-s − 0.219·113-s − 2.41·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 3.65·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.732·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{28}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{128} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{28} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.39195\)
\(L(\frac12)\)  \(\approx\)  \(2.39195\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 8. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
good3$C_2^2$ \( ( 1 + 22 p T^{2} + p^{8} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 1186 T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 1342 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 146 p^{2} T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 10466 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 162 T + p^{4} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 66242 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 62018 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 285854 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1453826 T^{2} + p^{8} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 1462178 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 1890 T + p^{4} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 1630462 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 7768706 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 11876386 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 19112066 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 22046306 T^{2} + p^{8} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 37495106 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 9393982 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 2750 T + p^{4} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 13977986 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 7728578 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 2430 T + p^{4} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 7454 T + p^{4} T^{2} )^{4} \)
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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

−9.102775311454724622978638898187, −8.868735491195685769555493975029, −8.713018086859630720791893717218, −8.303002650445643601614777583685, −7.84752100203802166133265794214, −7.72900218503414445701084150518, −7.46300456700746329920712342331, −6.87441255364502176155336622787, −6.66415729030919833489966812178, −6.38381933565016609506369513656, −6.31301053656432889173589406297, −5.78292837073453647806220662514, −5.48470306683953551851014093124, −4.90759398588364198377024704612, −4.88406142502632611400847446449, −4.58293645102061420242842711241, −3.96731406971884935726124658095, −3.77985337688437496605372513292, −3.02782882218380617815054436910, −2.79943439299283663261414756601, −2.48206631853757054646724900749, −2.23060336465473971407639389427, −1.29696649093180843465772607085, −0.78513212578891077436934625778, −0.37982586459999192134855018660, 0.37982586459999192134855018660, 0.78513212578891077436934625778, 1.29696649093180843465772607085, 2.23060336465473971407639389427, 2.48206631853757054646724900749, 2.79943439299283663261414756601, 3.02782882218380617815054436910, 3.77985337688437496605372513292, 3.96731406971884935726124658095, 4.58293645102061420242842711241, 4.88406142502632611400847446449, 4.90759398588364198377024704612, 5.48470306683953551851014093124, 5.78292837073453647806220662514, 6.31301053656432889173589406297, 6.38381933565016609506369513656, 6.66415729030919833489966812178, 6.87441255364502176155336622787, 7.46300456700746329920712342331, 7.72900218503414445701084150518, 7.84752100203802166133265794214, 8.303002650445643601614777583685, 8.713018086859630720791893717218, 8.868735491195685769555493975029, 9.102775311454724622978638898187

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