Properties

Label 8-80e8-1.1-c1e4-0-2
Degree $8$
Conductor $1.678\times 10^{15}$
Sign $1$
Analytic cond. $6.82069\times 10^{6}$
Root an. cond. $7.14872$
Motivic weight $1$
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  + 8·9-s + 48·41-s + 8·49-s + 30·81-s − 24·89-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 8/3·9-s + 7.49·41-s + 8/7·49-s + 10/3·81-s − 2.54·89-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.82069\times 10^{6}\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.329796544\)
\(L(\frac12)\) \(\approx\) \(9.329796544\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−5.70136602064571688255490547342, −5.39626410643701680754440301210, −5.33788310706385739388702613838, −4.97472278276881156017288804561, −4.94921603424922444341273411205, −4.50917411383501438840696031707, −4.39293310433670662386798343917, −4.25985471766429859919869643486, −4.18551643082977266123687688909, −4.05169645381293215502582831559, −3.79900489810019940404714660244, −3.67152415125918549243520950774, −3.43453838122108151308317996457, −2.85006167458386291820353692678, −2.83921206954150085600614730334, −2.57952931828557483201308391035, −2.54250479292172834031722608755, −2.21372874559264156248051810648, −2.01955599413479881017551878874, −1.49352982124345944233383597370, −1.45030015060695382885101537357, −1.22495267295920538543140061602, −0.894139317004329303341185546645, −0.75265696496809349060463928416, −0.35030165214176267515189230727, 0.35030165214176267515189230727, 0.75265696496809349060463928416, 0.894139317004329303341185546645, 1.22495267295920538543140061602, 1.45030015060695382885101537357, 1.49352982124345944233383597370, 2.01955599413479881017551878874, 2.21372874559264156248051810648, 2.54250479292172834031722608755, 2.57952931828557483201308391035, 2.83921206954150085600614730334, 2.85006167458386291820353692678, 3.43453838122108151308317996457, 3.67152415125918549243520950774, 3.79900489810019940404714660244, 4.05169645381293215502582831559, 4.18551643082977266123687688909, 4.25985471766429859919869643486, 4.39293310433670662386798343917, 4.50917411383501438840696031707, 4.94921603424922444341273411205, 4.97472278276881156017288804561, 5.33788310706385739388702613838, 5.39626410643701680754440301210, 5.70136602064571688255490547342

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