Properties

Label 8-1216e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 12·9-s − 14·17-s + 11·25-s + 24·45-s + 5·49-s + 30·61-s − 22·73-s + 90·81-s + 28·85-s + 40·101-s − 3·121-s − 38·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 168·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.894·5-s − 4·9-s − 3.39·17-s + 11/5·25-s + 3.57·45-s + 5/7·49-s + 3.84·61-s − 2.57·73-s + 10·81-s + 3.03·85-s + 3.98·101-s − 0.272·121-s − 3.39·125-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 + 13.5·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2536007541\)
\(L(\frac12)\) \(\approx\) \(0.2536007541\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{4} \)
5$C_2^2$ \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \)
11$C_2^2$$\times$$C_2^2$ \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} ) \)
47$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} )( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
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

−6.93169628705816716995807753944, −6.62695190903962523614635600408, −6.59112609498170056735028018995, −6.23638268251933107140580362478, −6.22874240249406335119383773593, −5.71527688299552490967522665542, −5.64201792171364101788997702964, −5.47996785995750485803002956296, −5.11068542833249976671561847951, −5.07908684880304459973525029547, −4.52393997958540853495981910957, −4.46217021859284210959720279501, −4.43211591071525013938648238598, −3.84917689826698829926305016667, −3.66776447146328898094091819872, −3.31322421222792031243241309871, −3.14026589385116810095749408704, −2.95131376303233461609682051183, −2.46721256464687153039851315276, −2.35283976722074823095843499678, −2.29995058278422064440349005770, −1.88869366439457487855130310765, −1.01095105835856636341406806106, −0.63074735473076003572080596253, −0.16788113180149400012379331689, 0.16788113180149400012379331689, 0.63074735473076003572080596253, 1.01095105835856636341406806106, 1.88869366439457487855130310765, 2.29995058278422064440349005770, 2.35283976722074823095843499678, 2.46721256464687153039851315276, 2.95131376303233461609682051183, 3.14026589385116810095749408704, 3.31322421222792031243241309871, 3.66776447146328898094091819872, 3.84917689826698829926305016667, 4.43211591071525013938648238598, 4.46217021859284210959720279501, 4.52393997958540853495981910957, 5.07908684880304459973525029547, 5.11068542833249976671561847951, 5.47996785995750485803002956296, 5.64201792171364101788997702964, 5.71527688299552490967522665542, 6.22874240249406335119383773593, 6.23638268251933107140580362478, 6.59112609498170056735028018995, 6.62695190903962523614635600408, 6.93169628705816716995807753944

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