Properties

Label 8-1200e4-1.1-c1e4-0-18
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $8430.11$
Root an. cond. $3.09548$
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  − 4·3-s + 6·9-s + 24·23-s + 4·27-s − 24·47-s + 28·49-s − 8·61-s − 96·69-s − 37·81-s + 24·83-s + 72·107-s + 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s − 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s + 5.00·23-s + 0.769·27-s − 3.50·47-s + 4·49-s − 1.02·61-s − 11.5·69-s − 4.11·81-s + 2.63·83-s + 6.96·107-s + 0.766·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s − 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8430.11\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.641535451\)
\(L(\frac12)\) \(\approx\) \(1.641535451\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−7.02837343258190653507048087634, −6.60699832504716808653534874994, −6.33167253430315769380062780932, −6.31285865637278628295550530124, −6.26481220095622331829477383714, −5.75036462864577452775225037478, −5.70583011448270234491638969920, −5.32013987318024694124460409520, −5.21093337067728124045211522524, −4.83301292442243581886206884061, −4.82042653783507017369988367564, −4.78466107278159807219195134980, −4.53496931017069318699897472810, −4.03814327840794976747259224922, −3.59102511912768982253783679702, −3.49414225528935483124573925259, −3.07623460161938996424843183634, −3.04213165138257558841128148061, −2.66869962165406348198435109199, −2.29069040869609445714004762243, −1.90662655269587847477109080815, −1.37448176811266162374016199869, −1.04553373541549380425500131675, −0.68888718498217436963196781908, −0.53832429936472359896102591639, 0.53832429936472359896102591639, 0.68888718498217436963196781908, 1.04553373541549380425500131675, 1.37448176811266162374016199869, 1.90662655269587847477109080815, 2.29069040869609445714004762243, 2.66869962165406348198435109199, 3.04213165138257558841128148061, 3.07623460161938996424843183634, 3.49414225528935483124573925259, 3.59102511912768982253783679702, 4.03814327840794976747259224922, 4.53496931017069318699897472810, 4.78466107278159807219195134980, 4.82042653783507017369988367564, 4.83301292442243581886206884061, 5.21093337067728124045211522524, 5.32013987318024694124460409520, 5.70583011448270234491638969920, 5.75036462864577452775225037478, 6.26481220095622331829477383714, 6.31285865637278628295550530124, 6.33167253430315769380062780932, 6.60699832504716808653534874994, 7.02837343258190653507048087634

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