Properties

Label 8-48e8-1.1-c2e4-0-18
Degree $8$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $1.55335\times 10^{7}$
Root an. cond. $7.92334$
Motivic weight $2$
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  + 92·25-s − 4·49-s + 200·73-s − 760·97-s − 284·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 3.67·25-s − 0.0816·49-s + 2.73·73-s − 7.83·97-s − 2.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.55335\times 10^{7}\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.072576293\)
\(L(\frac12)\) \(\approx\) \(4.072576293\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_2^2$ \( ( 1 + 818 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2^2$ \( ( 1 + 3218 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \)
61$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 9118 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 134 T + p^{2} T^{2} )^{2}( 1 + 134 T + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 190 T + p^{2} T^{2} )^{4} \)
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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

−6.43320588486298398285821881663, −5.89004258123237020347789566099, −5.82150250724435124139643067920, −5.52321175526001006889640584247, −5.20123061023789607131917113296, −5.15578815112845455471770408134, −5.11553320465664845746111863737, −4.82638039584402608219634442735, −4.48832426156109599192567547228, −4.22435810731441905349335624962, −4.09134338724035898720703239832, −3.94899011098152153020771940081, −3.56658379187541481137245249531, −3.38723878150185355565469140358, −3.01790591740906781322941013768, −2.77837665467781597611381693333, −2.74248085307893404781853927702, −2.47551979418242481886955964312, −2.25426701470460094592773443649, −1.50817404923469929958022019667, −1.49140814815396626427717561672, −1.44567942849851958235677034953, −0.74198211856186498862008881891, −0.72059049115024612794924143916, −0.25034872120441938495022209497, 0.25034872120441938495022209497, 0.72059049115024612794924143916, 0.74198211856186498862008881891, 1.44567942849851958235677034953, 1.49140814815396626427717561672, 1.50817404923469929958022019667, 2.25426701470460094592773443649, 2.47551979418242481886955964312, 2.74248085307893404781853927702, 2.77837665467781597611381693333, 3.01790591740906781322941013768, 3.38723878150185355565469140358, 3.56658379187541481137245249531, 3.94899011098152153020771940081, 4.09134338724035898720703239832, 4.22435810731441905349335624962, 4.48832426156109599192567547228, 4.82638039584402608219634442735, 5.11553320465664845746111863737, 5.15578815112845455471770408134, 5.20123061023789607131917113296, 5.52321175526001006889640584247, 5.82150250724435124139643067920, 5.89004258123237020347789566099, 6.43320588486298398285821881663

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