Properties

Label 8-96e8-1.1-c1e4-0-4
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
Motivic weight $1$
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  + 24·19-s + 4·25-s + 24·43-s − 24·49-s − 32·67-s − 48·73-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 5.50·19-s + 4/5·25-s + 3.65·43-s − 3.42·49-s − 3.90·67-s − 5.61·73-s + 4/11·121-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 − 3.69·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.805082287\)
\(L(\frac12)\) \(\approx\) \(4.805082287\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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

−5.54643703755115722252275865552, −5.32856324988424157066431688811, −5.10237112928620599902096303998, −4.74676703819523217533383360100, −4.73778748247084350393802217505, −4.52320016458681835185982467920, −4.30387988848262538151909509128, −4.15371331839441195151583207274, −4.12864062355283493116374798904, −3.60443294835984048565874149679, −3.33178875374962359603082037118, −3.24572694440190953289802574550, −3.15756537599390890860562648810, −3.01535763761885825616107218715, −2.83294256647400753416577295156, −2.70014187432636384264530679912, −2.49690078150222062003343782749, −1.90676579022112223803440717071, −1.71158503995434411321969205045, −1.49494183456744791130164007006, −1.48046839488398388983195045215, −1.09678853783157282449521512069, −0.811667604645753709043682570700, −0.71552529136574057574932915843, −0.22855338159461432523021291804, 0.22855338159461432523021291804, 0.71552529136574057574932915843, 0.811667604645753709043682570700, 1.09678853783157282449521512069, 1.48046839488398388983195045215, 1.49494183456744791130164007006, 1.71158503995434411321969205045, 1.90676579022112223803440717071, 2.49690078150222062003343782749, 2.70014187432636384264530679912, 2.83294256647400753416577295156, 3.01535763761885825616107218715, 3.15756537599390890860562648810, 3.24572694440190953289802574550, 3.33178875374962359603082037118, 3.60443294835984048565874149679, 4.12864062355283493116374798904, 4.15371331839441195151583207274, 4.30387988848262538151909509128, 4.52320016458681835185982467920, 4.73778748247084350393802217505, 4.74676703819523217533383360100, 5.10237112928620599902096303998, 5.32856324988424157066431688811, 5.54643703755115722252275865552

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