Properties

Label 8-1792e4-1.1-c1e4-0-19
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $41923.7$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 4·9-s − 8·17-s − 16·23-s − 12·25-s − 8·41-s − 32·47-s + 10·49-s + 16·63-s − 32·71-s − 24·73-s − 32·79-s + 2·81-s + 8·89-s − 40·97-s − 32·103-s + 24·113-s + 32·119-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + ⋯
L(s)  = 1  − 1.51·7-s − 4/3·9-s − 1.94·17-s − 3.33·23-s − 2.39·25-s − 1.24·41-s − 4.66·47-s + 10/7·49-s + 2.01·63-s − 3.79·71-s − 2.80·73-s − 3.60·79-s + 2/9·81-s + 0.847·89-s − 4.06·97-s − 3.15·103-s + 2.25·113-s + 2.93·119-s − 1.09·121-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 + 2.58·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(41923.7\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1792} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
good3$C_2^2:C_4$ \( 1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^2:C_4$ \( 1 + 12 T^{2} + 78 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 + 12 T^{2} + 150 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 + 12 T^{2} - 18 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_4$ \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2:C_4$ \( 1 + 36 T^{2} + 1038 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 - 44 T^{2} + 2038 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2:C_4$ \( 1 + 116 T^{2} + 5974 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2:C_4$ \( 1 + 140 T^{2} + 8470 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
53$C_2^2:C_4$ \( 1 + 84 T^{2} + 5334 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 + 196 T^{2} + 16174 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 + 44 T^{2} + 5614 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 + 204 T^{2} + 18870 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2:C_4$ \( 1 + 132 T^{2} + 13134 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−6.89776450129221367628295002733, −6.71024082674822773634258999550, −6.65760381391867760818511185246, −6.26140831341748450815939759883, −6.10806874249039755051557367326, −6.07336050326859306649643219852, −5.73509068248932338239097733842, −5.68139668489037843911617459373, −5.59326452443745126862025413529, −5.15030013912565917060201334438, −4.76013177949930526529601991251, −4.60227341785372477694002109423, −4.30748513064606110658485007892, −4.29496806204231556646243484392, −3.80052614364130295761363185210, −3.71720930541563799753300226314, −3.68457316220561765646885140016, −2.94079090463619968806675059787, −2.92941396839128490350689486547, −2.88224036748915104976257351764, −2.58555944949859238717739070058, −2.03196844387901075673338940956, −1.80902780622613050753272117026, −1.55268222142430473184260242933, −1.50615838321679419064906625240, 0, 0, 0, 0, 1.50615838321679419064906625240, 1.55268222142430473184260242933, 1.80902780622613050753272117026, 2.03196844387901075673338940956, 2.58555944949859238717739070058, 2.88224036748915104976257351764, 2.92941396839128490350689486547, 2.94079090463619968806675059787, 3.68457316220561765646885140016, 3.71720930541563799753300226314, 3.80052614364130295761363185210, 4.29496806204231556646243484392, 4.30748513064606110658485007892, 4.60227341785372477694002109423, 4.76013177949930526529601991251, 5.15030013912565917060201334438, 5.59326452443745126862025413529, 5.68139668489037843911617459373, 5.73509068248932338239097733842, 6.07336050326859306649643219852, 6.10806874249039755051557367326, 6.26140831341748450815939759883, 6.65760381391867760818511185246, 6.71024082674822773634258999550, 6.89776450129221367628295002733

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