Properties

Label 8-4032e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.643\times 10^{14}$
Sign $1$
Analytic cond. $1.07446\times 10^{6}$
Root an. cond. $5.67412$
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  − 4·7-s + 12·17-s − 28·23-s + 12·25-s − 8·31-s − 12·41-s + 10·49-s − 36·71-s − 24·79-s + 12·89-s − 16·97-s − 8·103-s − 48·113-s − 48·119-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 112·161-s + 163-s + 167-s + 52·169-s + 173-s + ⋯
L(s)  = 1  − 1.51·7-s + 2.91·17-s − 5.83·23-s + 12/5·25-s − 1.43·31-s − 1.87·41-s + 10/7·49-s − 4.27·71-s − 2.70·79-s + 1.27·89-s − 1.62·97-s − 0.788·103-s − 4.51·113-s − 4.40·119-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 + 0.0798·157-s + 8.82·161-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.07446\times 10^{6}\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.006600572555\)
\(L(\frac12)\) \(\approx\) \(0.006600572555\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
good5$D_4\times C_2$ \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 12 T^{2} + 170 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$D_{4}$ \( ( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 14 T + 4 p T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4 T^{2} - 3210 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
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$D_4\times C_2$ \( 1 - 108 T^{2} + 6806 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 212 T^{2} + 18486 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 212 T^{2} + 19446 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 108 T^{2} + 14966 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 6 T + 184 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.90015065226944711634362097233, −5.70440004506850664796171822565, −5.56353371920335253501831718267, −5.47673400868102769504977463381, −5.33529163438221789157941074702, −5.22564889239398275631954240486, −4.73057092403956957505865346557, −4.30174136580657657011241273236, −4.22560862067560772269646216458, −4.20391959776076486716993797813, −4.05423167429881043697774639732, −3.63725387314756800121681071508, −3.61268770071791628249959436620, −3.12820459803280887467756092681, −3.03962282337937458567069250130, −2.97723232323950616382614542180, −2.87019133457855893607214931238, −2.24156970388928448736207447019, −2.16926655229267259649830700377, −1.77444296562930855913048072626, −1.46888842529154460647010409069, −1.45593759879866825597827626227, −1.04859275969819145266774259185, −0.42258028664382903963560359691, −0.01724161982355531135631982883, 0.01724161982355531135631982883, 0.42258028664382903963560359691, 1.04859275969819145266774259185, 1.45593759879866825597827626227, 1.46888842529154460647010409069, 1.77444296562930855913048072626, 2.16926655229267259649830700377, 2.24156970388928448736207447019, 2.87019133457855893607214931238, 2.97723232323950616382614542180, 3.03962282337937458567069250130, 3.12820459803280887467756092681, 3.61268770071791628249959436620, 3.63725387314756800121681071508, 4.05423167429881043697774639732, 4.20391959776076486716993797813, 4.22560862067560772269646216458, 4.30174136580657657011241273236, 4.73057092403956957505865346557, 5.22564889239398275631954240486, 5.33529163438221789157941074702, 5.47673400868102769504977463381, 5.56353371920335253501831718267, 5.70440004506850664796171822565, 5.90015065226944711634362097233

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