Properties

Degree $8$
Conductor $9.683\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 8·8-s − 4·16-s + 14·25-s − 16·29-s + 32·32-s + 12·37-s − 56·50-s + 4·53-s + 64·58-s − 64·64-s − 48·74-s + 112·100-s − 16·106-s + 36·109-s + 64·113-s − 128·116-s + 42·121-s + 127-s + 64·128-s + 131-s + 137-s + 139-s + 96·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 2.82·8-s − 16-s + 14/5·25-s − 2.97·29-s + 5.65·32-s + 1.97·37-s − 7.91·50-s + 0.549·53-s + 8.40·58-s − 8·64-s − 5.57·74-s + 56/5·100-s − 1.55·106-s + 3.44·109-s + 6.02·113-s − 11.8·116-s + 3.81·121-s + 0.0887·127-s + 5.65·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.89·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5918901884\)
\(L(\frac12)\) \(\approx\) \(0.5918901884\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.84829286702993494992917504166, −6.54275645516538734442746771850, −6.32228063502552642530878523169, −5.99528243713056950803383651794, −5.92022651526108284663064294371, −5.68456782931109957100422552617, −5.56114543391741238765336201377, −4.88980461809374284139112356957, −4.75662625396817578646449208347, −4.74156242101763423484470598179, −4.71189647367414077419706756442, −4.08393409131228156063404608575, −4.06146484947744979374346291766, −3.58333101498185773125205387877, −3.35769054750328220706337392777, −3.10004910286515755511790684133, −2.91149061571937279277692471832, −2.33095361418293132161863258608, −2.27290889800607083619168144811, −1.89323360354094045278905221403, −1.87817368842723006370202751786, −1.26575532208995374702107014886, −0.872182168283802581494826426066, −0.835315931188436297924140425206, −0.32154438462153693801956957110, 0.32154438462153693801956957110, 0.835315931188436297924140425206, 0.872182168283802581494826426066, 1.26575532208995374702107014886, 1.87817368842723006370202751786, 1.89323360354094045278905221403, 2.27290889800607083619168144811, 2.33095361418293132161863258608, 2.91149061571937279277692471832, 3.10004910286515755511790684133, 3.35769054750328220706337392777, 3.58333101498185773125205387877, 4.06146484947744979374346291766, 4.08393409131228156063404608575, 4.71189647367414077419706756442, 4.74156242101763423484470598179, 4.75662625396817578646449208347, 4.88980461809374284139112356957, 5.56114543391741238765336201377, 5.68456782931109957100422552617, 5.92022651526108284663064294371, 5.99528243713056950803383651794, 6.32228063502552642530878523169, 6.54275645516538734442746771850, 6.84829286702993494992917504166

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