Properties

Label 8-50e8-1.1-c0e4-0-5
Degree $8$
Conductor $3.906\times 10^{13}$
Sign $1$
Analytic cond. $2.42319$
Root an. cond. $1.11698$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 9-s + 3·13-s + 3·17-s + 18-s − 3·26-s + 3·29-s + 32-s − 3·34-s − 2·37-s + 3·41-s + 4·49-s − 2·53-s − 3·58-s + 3·61-s − 64-s + 3·73-s + 2·74-s − 3·82-s − 2·89-s + 3·97-s − 4·98-s − 2·101-s + 2·106-s + 3·109-s − 2·113-s − 3·117-s + ⋯
L(s)  = 1  − 2-s − 9-s + 3·13-s + 3·17-s + 18-s − 3·26-s + 3·29-s + 32-s − 3·34-s − 2·37-s + 3·41-s + 4·49-s − 2·53-s − 3·58-s + 3·61-s − 64-s + 3·73-s + 2·74-s − 3·82-s − 2·89-s + 3·97-s − 4·98-s − 2·101-s + 2·106-s + 3·109-s − 2·113-s − 3·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(2.42319\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186706910\)
\(L(\frac12)\) \(\approx\) \(1.186706910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5 \( 1 \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.63291453588320583816204477210, −6.15227828822242522152735180767, −6.12904923794926219593187198082, −5.92656706472065396111437987045, −5.73440878559893036806229518181, −5.54300650437359506044810565373, −5.50362124000553685207637829266, −5.14382458355951634825515537200, −4.99060614554309244578703277541, −4.50679504468833932241103769960, −4.50581275880461499129304215114, −4.11864284953562135152126630277, −3.86946682190427220348141451577, −3.71189609442901170933963957361, −3.52906757319584919597598803973, −3.16504215310281493827556082027, −3.11328672026137483943424580955, −2.95374727689042182857232603560, −2.26680092853660939200980642338, −2.25898291840861027747833050121, −2.23444745983071046506044731110, −1.18976783125266095511919614404, −1.07378338559188825633651364852, −1.06815228181934673297886953534, −0.900456715950755990029111354991, 0.900456715950755990029111354991, 1.06815228181934673297886953534, 1.07378338559188825633651364852, 1.18976783125266095511919614404, 2.23444745983071046506044731110, 2.25898291840861027747833050121, 2.26680092853660939200980642338, 2.95374727689042182857232603560, 3.11328672026137483943424580955, 3.16504215310281493827556082027, 3.52906757319584919597598803973, 3.71189609442901170933963957361, 3.86946682190427220348141451577, 4.11864284953562135152126630277, 4.50581275880461499129304215114, 4.50679504468833932241103769960, 4.99060614554309244578703277541, 5.14382458355951634825515537200, 5.50362124000553685207637829266, 5.54300650437359506044810565373, 5.73440878559893036806229518181, 5.92656706472065396111437987045, 6.12904923794926219593187198082, 6.15227828822242522152735180767, 6.63291453588320583816204477210

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