Properties

Label 8-2352e4-1.1-c1e4-0-13
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
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 + 10·9-s + 16·25-s + 20·27-s − 16·29-s − 16·31-s + 16·47-s − 16·53-s + 64·75-s + 35·81-s + 48·83-s − 64·87-s − 64·93-s + 48·103-s + 32·109-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 64·141-s + 149-s + 151-s + 157-s − 64·159-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 16/5·25-s + 3.84·27-s − 2.97·29-s − 2.87·31-s + 2.33·47-s − 2.19·53-s + 7.39·75-s + 35/9·81-s + 5.26·83-s − 6.86·87-s − 6.63·93-s + 4.72·103-s + 3.06·109-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.38·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 5.07·159-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.14179398\)
\(L(\frac12)\) \(\approx\) \(11.14179398\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
good5$C_2^2:C_4$ \( 1 - 16 T^{2} + 112 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
17$C_4\times C_2$ \( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2:C_4$ \( 1 - 60 T^{2} + 1830 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2:C_4$ \( 1 - 112 T^{2} + 6400 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2:C_4$ \( 1 - 108 T^{2} + 6102 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2:C_4$ \( 1 - 192 T^{2} + 16080 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 + 52 T^{2} + 9142 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 4 T^{2} - 282 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 224 T^{2} + 23104 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 + 4 T^{2} + 11974 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2:C_4$ \( 1 - 144 T^{2} + 20448 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.40558395259215105392406140932, −6.16177882530729887930846486521, −5.94519973572594769446563389949, −5.79649754822801686159012134848, −5.66801676917504194058890372433, −5.07018134389712325351750846340, −5.00792745647270812800050564428, −4.94869754525340680025568071171, −4.76492584292363527288067393805, −4.36156762003876146327930896838, −4.21203674164188545863277797018, −3.83092870778548275030684711555, −3.69245251718891176062364560620, −3.44998011274917484814197161972, −3.29554408947505564236705813436, −3.15716297602895427291552467882, −3.07080550574221265846968830216, −2.46434891327894550737317990236, −2.10495074880529270675042426608, −2.08211603482157431766623014672, −2.07132934076936626575244825509, −1.66888168873993637368326272845, −1.09277059009923260306692563502, −0.925156714997007642044198789535, −0.41651503844867982351506706796, 0.41651503844867982351506706796, 0.925156714997007642044198789535, 1.09277059009923260306692563502, 1.66888168873993637368326272845, 2.07132934076936626575244825509, 2.08211603482157431766623014672, 2.10495074880529270675042426608, 2.46434891327894550737317990236, 3.07080550574221265846968830216, 3.15716297602895427291552467882, 3.29554408947505564236705813436, 3.44998011274917484814197161972, 3.69245251718891176062364560620, 3.83092870778548275030684711555, 4.21203674164188545863277797018, 4.36156762003876146327930896838, 4.76492584292363527288067393805, 4.94869754525340680025568071171, 5.00792745647270812800050564428, 5.07018134389712325351750846340, 5.66801676917504194058890372433, 5.79649754822801686159012134848, 5.94519973572594769446563389949, 6.16177882530729887930846486521, 6.40558395259215105392406140932

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