Properties

Degree $8$
Conductor $1.216\times 10^{12}$
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  + 4-s + 9-s + 10·11-s − 14·19-s + 4·31-s + 36-s + 20·41-s + 10·44-s − 2·49-s + 8·59-s − 8·61-s − 64-s + 8·71-s − 14·76-s − 24·79-s + 28·89-s + 10·99-s − 4·109-s + 47·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s + 3.01·11-s − 3.21·19-s + 0.718·31-s + 1/6·36-s + 3.12·41-s + 1.50·44-s − 2/7·49-s + 1.04·59-s − 1.02·61-s − 1/8·64-s + 0.949·71-s − 1.60·76-s − 2.70·79-s + 2.96·89-s + 1.00·99-s − 0.383·109-s + 4.27·121-s + 0.359·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.956585065\)
\(L(\frac12)\) \(\approx\) \(4.956585065\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
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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

−6.97991061745274308466963357169, −6.82497705710978033604737071180, −6.52975980915240054040325065528, −6.38985338402122664232791814522, −6.35651595112453648970859603778, −6.15343457832562239082049460922, −5.78067556119195978892506353556, −5.77387077129675827100604285956, −5.26837170203719485089935673111, −5.01310478053945939279262613111, −4.60876521342614711698159819409, −4.34395517083850080423882480809, −4.22769619086154226136444841032, −4.17549894720672970958948644516, −3.97641672944950888662889964838, −3.59178548527415140290518200655, −3.22710904009703522802973089291, −3.04098226914475737464000653999, −2.49987477204592274667071763532, −2.34152392308899562247198033601, −2.03869198244360550765590247557, −1.64621132292252629200048457183, −1.47944023192445944701519406839, −0.881726754073095681531316022913, −0.56281755428133394351010228655, 0.56281755428133394351010228655, 0.881726754073095681531316022913, 1.47944023192445944701519406839, 1.64621132292252629200048457183, 2.03869198244360550765590247557, 2.34152392308899562247198033601, 2.49987477204592274667071763532, 3.04098226914475737464000653999, 3.22710904009703522802973089291, 3.59178548527415140290518200655, 3.97641672944950888662889964838, 4.17549894720672970958948644516, 4.22769619086154226136444841032, 4.34395517083850080423882480809, 4.60876521342614711698159819409, 5.01310478053945939279262613111, 5.26837170203719485089935673111, 5.77387077129675827100604285956, 5.78067556119195978892506353556, 6.15343457832562239082049460922, 6.35651595112453648970859603778, 6.38985338402122664232791814522, 6.52975980915240054040325065528, 6.82497705710978033604737071180, 6.97991061745274308466963357169

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