Properties

Label 8-96e8-1.1-c1e4-0-7
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·11-s − 12·19-s − 8·25-s − 12·43-s − 12·49-s + 36·59-s − 4·67-s − 24·73-s + 12·83-s + 24·89-s + 12·107-s + 24·113-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 3.61·11-s − 2.75·19-s − 8/5·25-s − 1.82·43-s − 1.71·49-s + 4.68·59-s − 0.488·67-s − 2.80·73-s + 1.31·83-s + 2.54·89-s + 1.16·107-s + 2.25·113-s + 4.72·121-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 − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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: Trivial
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

\(L(1)\) \(\approx\) \(5.364236877\)
\(L(\frac12)\) \(\approx\) \(5.364236877\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 + 12 T^{2} + 86 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 24 T^{2} + 290 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 44 T^{2} + 1110 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 120 T^{2} + 6146 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 56 T^{2} + 4674 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 216 T^{2} + 18914 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 236 T^{2} + 23574 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 60 T^{2} + 1094 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 182 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.64494925088103862315255689600, −5.01903082420592550042201311864, −4.98997898484352017343956703680, −4.79770544477028093348548403550, −4.77184564248947787093814186576, −4.26112353198806899539648154890, −4.25523828952552768741029028962, −4.17159595337614414442709105622, −4.04711833134030442918064431698, −3.59494180004719624351520774074, −3.59304366012667201950865406160, −3.47561563471115102263467808067, −3.44304185538197240726352333669, −2.96025346584817678945712420233, −2.64013453819986174981104322551, −2.49425333108933650414003467380, −2.24278278189946080191653728673, −1.83411330880346049545336483234, −1.81948209515805917387635077235, −1.70712505690174804079330533861, −1.60626829290271172056419249974, −0.998720882648904136771165712044, −0.929223168791297267810133906094, −0.48928402475180207401501095773, −0.30267054945420398874996313828, 0.30267054945420398874996313828, 0.48928402475180207401501095773, 0.929223168791297267810133906094, 0.998720882648904136771165712044, 1.60626829290271172056419249974, 1.70712505690174804079330533861, 1.81948209515805917387635077235, 1.83411330880346049545336483234, 2.24278278189946080191653728673, 2.49425333108933650414003467380, 2.64013453819986174981104322551, 2.96025346584817678945712420233, 3.44304185538197240726352333669, 3.47561563471115102263467808067, 3.59304366012667201950865406160, 3.59494180004719624351520774074, 4.04711833134030442918064431698, 4.17159595337614414442709105622, 4.25523828952552768741029028962, 4.26112353198806899539648154890, 4.77184564248947787093814186576, 4.79770544477028093348548403550, 4.98997898484352017343956703680, 5.01903082420592550042201311864, 5.64494925088103862315255689600

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