Properties

Label 8-50e8-1.1-c0e4-0-6
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 2·6-s + 2·7-s + 9-s + 2·14-s + 18-s + 4·21-s + 2·23-s − 2·29-s − 32-s + 3·41-s + 4·42-s + 2·43-s + 2·46-s + 2·47-s + 49-s − 2·58-s − 2·61-s + 2·63-s − 64-s + 2·67-s + 4·69-s + 3·82-s − 3·83-s + 2·86-s − 4·87-s + ⋯
L(s)  = 1  + 2-s + 2·3-s + 2·6-s + 2·7-s + 9-s + 2·14-s + 18-s + 4·21-s + 2·23-s − 2·29-s − 32-s + 3·41-s + 4·42-s + 2·43-s + 2·46-s + 2·47-s + 49-s − 2·58-s − 2·61-s + 2·63-s − 64-s + 2·67-s + 4·69-s + 3·82-s − 3·83-s + 2·86-s − 4·87-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

\(L(\frac{1}{2})\) \(\approx\) \(7.025181663\)
\(L(\frac12)\) \(\approx\) \(7.025181663\)
\(L(1)\) 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$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
7$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
29$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
47$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.42213669012807378572691820264, −6.19810842539751469935907022040, −6.04249410243797351577465301196, −5.79360395593900857721083594973, −5.57457571373889726278445499898, −5.47647887680163473249673558240, −5.26913128329759864484970343098, −5.01359347097360965261668888613, −4.74010947108876759133897983603, −4.56780752508732923047161270625, −4.46203862343632856159862758270, −4.09924138041649695139441526880, −4.03920749211525859179218939814, −3.88636518146787945192765184401, −3.55155787620817657507054437706, −3.18877364614556452628517955038, −3.04189233844060612608802731490, −2.84018867329524142305940854648, −2.49823782775422098636233583311, −2.47492414731401688452165838901, −2.00617053881978415967273458275, −2.00222429803606701063238480600, −1.44181257772867618342750506408, −1.13390886438567413976883468019, −0.960225447932536745197182587749, 0.960225447932536745197182587749, 1.13390886438567413976883468019, 1.44181257772867618342750506408, 2.00222429803606701063238480600, 2.00617053881978415967273458275, 2.47492414731401688452165838901, 2.49823782775422098636233583311, 2.84018867329524142305940854648, 3.04189233844060612608802731490, 3.18877364614556452628517955038, 3.55155787620817657507054437706, 3.88636518146787945192765184401, 4.03920749211525859179218939814, 4.09924138041649695139441526880, 4.46203862343632856159862758270, 4.56780752508732923047161270625, 4.74010947108876759133897983603, 5.01359347097360965261668888613, 5.26913128329759864484970343098, 5.47647887680163473249673558240, 5.57457571373889726278445499898, 5.79360395593900857721083594973, 6.04249410243797351577465301196, 6.19810842539751469935907022040, 6.42213669012807378572691820264

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