Properties

Label 8-50e8-1.1-c0e4-0-3
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

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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: induced by $\chi_{2500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.529525769\)
\(L(\frac12)\) \(\approx\) \(1.529525769\)
\(L(1)\) 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

−6.46895173107416869387732625773, −6.29719705657601986265894002770, −6.14925205868528148928978132087, −5.63238755456889284065269091451, −5.54085708956403222787278723645, −5.52362361463056173792463037062, −5.51134884825536185985176763639, −4.88963896468221950264321865224, −4.76990885636083203532849094261, −4.67505230967454342017837014212, −4.45178045845089100633717738064, −4.09836404837452125144220482350, −4.08078958206713210802183495819, −4.02780278679253793547775505645, −3.92753396762493846777370611865, −2.91495901844413406656902309694, −2.88275204725785428114534985480, −2.80374193750276773034243885836, −2.66874459260068934211052937482, −2.39227651126043525990292547829, −2.13063426063701713359973488934, −2.08711285937985527778426775329, −1.38158039865059343066219962935, −0.69999218743651607262078314993, −0.67495917914578661225401303381, 0.67495917914578661225401303381, 0.69999218743651607262078314993, 1.38158039865059343066219962935, 2.08711285937985527778426775329, 2.13063426063701713359973488934, 2.39227651126043525990292547829, 2.66874459260068934211052937482, 2.80374193750276773034243885836, 2.88275204725785428114534985480, 2.91495901844413406656902309694, 3.92753396762493846777370611865, 4.02780278679253793547775505645, 4.08078958206713210802183495819, 4.09836404837452125144220482350, 4.45178045845089100633717738064, 4.67505230967454342017837014212, 4.76990885636083203532849094261, 4.88963896468221950264321865224, 5.51134884825536185985176763639, 5.52362361463056173792463037062, 5.54085708956403222787278723645, 5.63238755456889284065269091451, 6.14925205868528148928978132087, 6.29719705657601986265894002770, 6.46895173107416869387732625773

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