Properties

Degree 8
Conductor $ 2^{16} \cdot 3^{8} \cdot 7^{4} $
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  + 10·7-s + 8·19-s + 5·25-s − 2·31-s − 8·37-s + 61·49-s + 36·61-s − 24·67-s − 24·73-s + 42·79-s + 16·103-s + 8·109-s − 7·121-s + 127-s + 131-s + 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 50·175-s + 179-s + ⋯
L(s)  = 1  + 3.77·7-s + 1.83·19-s + 25-s − 0.359·31-s − 1.31·37-s + 61/7·49-s + 4.60·61-s − 2.93·67-s − 2.80·73-s + 4.72·79-s + 1.57·103-s + 0.766·109-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + 6.93·133-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 + 2.15·169-s + 0.0760·173-s + 3.77·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(7.792187594\)
\(L(\frac12)\)  \(\approx\)  \(7.792187594\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
good5$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 + 7 T^{2} - 72 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 14 T^{2} - 333 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 86 T^{2} + 5187 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 61 T^{2} + 912 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 73 T^{2} + 1848 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 18 T + 169 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 118 T^{2} + 6003 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.37619543517435020569116353335, −7.10227121930539293988648035325, −6.71493363026460909838599242864, −6.48814240683922879799795106919, −6.24160943624386512154633614810, −5.92297430354469545985470702092, −5.55454660153073689525482995804, −5.43928110111558380197361965484, −5.31310557199032734963821914368, −4.94654813181701953510236998979, −4.81877832650278341383899386361, −4.80952471233036566344778781263, −4.56399411658811557480021688914, −4.03226309903881700495659169597, −3.72506842785513394010179512699, −3.70426249213357988878858868000, −3.48222220852636701031637167404, −2.72337014160572371331039918924, −2.65570274181617204534961235923, −2.36129205700071365374079727117, −1.96455769706301333888722050995, −1.64274280156998604881553889558, −1.34475117445146891390206886479, −1.10657997650339053144143930908, −0.67496184238300226305476616076, 0.67496184238300226305476616076, 1.10657997650339053144143930908, 1.34475117445146891390206886479, 1.64274280156998604881553889558, 1.96455769706301333888722050995, 2.36129205700071365374079727117, 2.65570274181617204534961235923, 2.72337014160572371331039918924, 3.48222220852636701031637167404, 3.70426249213357988878858868000, 3.72506842785513394010179512699, 4.03226309903881700495659169597, 4.56399411658811557480021688914, 4.80952471233036566344778781263, 4.81877832650278341383899386361, 4.94654813181701953510236998979, 5.31310557199032734963821914368, 5.43928110111558380197361965484, 5.55454660153073689525482995804, 5.92297430354469545985470702092, 6.24160943624386512154633614810, 6.48814240683922879799795106919, 6.71493363026460909838599242864, 7.10227121930539293988648035325, 7.37619543517435020569116353335

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