Properties

Label 8-80e8-1.1-c1e4-0-1
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  + 8·11-s + 24·19-s + 16·41-s − 24·49-s + 8·59-s − 18·81-s − 8·89-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 192·209-s + 211-s + ⋯
L(s)  = 1  + 2.41·11-s + 5.50·19-s + 2.49·41-s − 3.42·49-s + 1.04·59-s − 2·81-s − 0.847·89-s − 0.363·121-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 + 0.923·169-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 + 13.2·209-s + 0.0688·211-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

\(L(1)\) \(\approx\) \(9.812664200\)
\(L(\frac12)\) \(\approx\) \(9.812664200\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.77764175008513907578167105030, −5.33191078382420866176656618962, −5.24140962882105301823378665285, −5.09383578230924223767749892298, −5.05933051368572319844058609918, −4.70295489216631238853998080135, −4.52346036167028144312980411102, −4.22029470388050458047247774273, −4.08139348629431370905245385777, −3.84387565032814262829180104915, −3.78441862009535141334696670362, −3.42231728069980766392072748944, −3.39182499684607108415342153288, −3.05137992931197684092436028341, −2.84512209145049885887062929233, −2.79751932829467436319646808086, −2.69296245190271906233102960119, −2.10787870166669289201335371429, −1.70878828299397121030804168063, −1.69002825336140590899147738754, −1.35265479541562967340386683530, −1.21509833642633123088956743928, −0.886363380029753462392263620790, −0.845458985825921989627349439376, −0.34144966940325296132723051137, 0.34144966940325296132723051137, 0.845458985825921989627349439376, 0.886363380029753462392263620790, 1.21509833642633123088956743928, 1.35265479541562967340386683530, 1.69002825336140590899147738754, 1.70878828299397121030804168063, 2.10787870166669289201335371429, 2.69296245190271906233102960119, 2.79751932829467436319646808086, 2.84512209145049885887062929233, 3.05137992931197684092436028341, 3.39182499684607108415342153288, 3.42231728069980766392072748944, 3.78441862009535141334696670362, 3.84387565032814262829180104915, 4.08139348629431370905245385777, 4.22029470388050458047247774273, 4.52346036167028144312980411102, 4.70295489216631238853998080135, 5.05933051368572319844058609918, 5.09383578230924223767749892298, 5.24140962882105301823378665285, 5.33191078382420866176656618962, 5.77764175008513907578167105030

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