Properties

Degree 8
Conductor $ 2^{24} \cdot 5^{4} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 972·9-s + 6.25e3·25-s + 6.08e4·41-s − 5.37e4·49-s + 5.90e5·81-s + 5.15e5·89-s − 6.42e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.36e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 6.07e6·225-s + ⋯
L(s)  = 1  − 4·9-s + 2·25-s + 5.64·41-s − 3.19·49-s + 10·81-s + 6.89·89-s − 3.98·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 3.67·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s − 8·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{24} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(1.508028091\)
\(L(\frac12)\)  \(\approx\)  \(1.508028091\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
good3$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
7$C_2^2$ \( ( 1 + 26886 T^{2} + p^{10} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 321102 T^{2} + p^{10} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 682086 T^{2} + p^{10} T^{4} )^{2} \)
17$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 1288802 T^{2} + p^{10} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 1772186 T^{2} + p^{10} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
31$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 112652586 T^{2} + p^{10} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 15202 T + p^{5} T^{2} )^{4} \)
43$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 222705514 T^{2} + p^{10} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 552886486 T^{2} + p^{10} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 646632402 T^{2} + p^{10} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
67$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
71$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
73$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
79$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
83$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
89$C_2$ \( ( 1 - 128786 T + p^{5} T^{2} )^{4} \)
97$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
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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

−7.76183234829800881270918482446, −7.42923494378033084855033595962, −7.19441764679067891009491935754, −6.48234610938060760380199711343, −6.41801082014443549506739662067, −6.23810048339743478371387697424, −6.05007141337034201631350484736, −5.92976261812794858455445699862, −5.44310181677161799662068178105, −5.11322377232845712621327229034, −5.01186322965047259224628202810, −4.86691942303921970454520533169, −4.40927904992655342356013119282, −3.84282787951566465870555092599, −3.72183536243922442378323533308, −3.27258461960101866368017691321, −2.97390815804244769131848872014, −2.82040731214383640035707393217, −2.52865423711880795406053004803, −2.27675507796219162943257077935, −1.94739011034032066857403606604, −1.11128989406410684344367236482, −0.883094249850789211345297601727, −0.57299733356626466473965749556, −0.20895829852626990355225425014, 0.20895829852626990355225425014, 0.57299733356626466473965749556, 0.883094249850789211345297601727, 1.11128989406410684344367236482, 1.94739011034032066857403606604, 2.27675507796219162943257077935, 2.52865423711880795406053004803, 2.82040731214383640035707393217, 2.97390815804244769131848872014, 3.27258461960101866368017691321, 3.72183536243922442378323533308, 3.84282787951566465870555092599, 4.40927904992655342356013119282, 4.86691942303921970454520533169, 5.01186322965047259224628202810, 5.11322377232845712621327229034, 5.44310181677161799662068178105, 5.92976261812794858455445699862, 6.05007141337034201631350484736, 6.23810048339743478371387697424, 6.41801082014443549506739662067, 6.48234610938060760380199711343, 7.19441764679067891009491935754, 7.42923494378033084855033595962, 7.76183234829800881270918482446

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