Properties

Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·9-s − 6·11-s + 2·19-s − 8·29-s + 10·31-s − 8·41-s − 2·49-s + 30·59-s + 10·61-s + 2·79-s + 9·81-s + 14·89-s + 30·99-s − 6·101-s − 10·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s − 10·171-s + ⋯
L(s)  = 1  − 5/3·9-s − 1.80·11-s + 0.458·19-s − 1.48·29-s + 1.79·31-s − 1.24·41-s − 2/7·49-s + 3.90·59-s + 1.28·61-s + 0.225·79-s + 81-s + 1.48·89-s + 3.01·99-s − 0.597·101-s − 0.957·109-s + 2.81·121-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 − 1.53·169-s − 0.764·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3245539403\)
\(L(\frac12)\) \(\approx\) \(0.3245539403\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^3$ \( 1 + 9 T^{2} - 208 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 69 T^{2} + 2552 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.76874181712498439937589566431, −6.70057596776158568517647903846, −6.40972783740450337227118053151, −5.93926047337008311897925882340, −5.89742857156455839942674100486, −5.87592895374555893902510125540, −5.35604468720461207669288080531, −5.33276902726813355436507377495, −5.15933199347355835271774348098, −4.96784455283808950449597526500, −4.64353473214225688645223870650, −4.49627071815451384743796376604, −3.90822166245208246362551537234, −3.85941950528837757437702710607, −3.61853060528631916355184799554, −3.31526122371609891395444245742, −3.02976048155125047227069162442, −2.75169211778699393149820739983, −2.60157821688622295031232664876, −2.23570270255096852800178402951, −2.11373911164623239737350057742, −1.74988872079546523244696139985, −0.962581114796030014492131240450, −0.883185381773031845670721990047, −0.13475906297086189953520179102, 0.13475906297086189953520179102, 0.883185381773031845670721990047, 0.962581114796030014492131240450, 1.74988872079546523244696139985, 2.11373911164623239737350057742, 2.23570270255096852800178402951, 2.60157821688622295031232664876, 2.75169211778699393149820739983, 3.02976048155125047227069162442, 3.31526122371609891395444245742, 3.61853060528631916355184799554, 3.85941950528837757437702710607, 3.90822166245208246362551537234, 4.49627071815451384743796376604, 4.64353473214225688645223870650, 4.96784455283808950449597526500, 5.15933199347355835271774348098, 5.33276902726813355436507377495, 5.35604468720461207669288080531, 5.87592895374555893902510125540, 5.89742857156455839942674100486, 5.93926047337008311897925882340, 6.40972783740450337227118053151, 6.70057596776158568517647903846, 6.76874181712498439937589566431

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