Properties

Label 8-273e4-1.1-c1e4-0-5
Degree $8$
Conductor $5554571841$
Sign $1$
Analytic cond. $22.5818$
Root an. cond. $1.47645$
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  + 3·2-s + 4·3-s + 3·4-s − 6·5-s + 12·6-s + 10·9-s − 18·10-s + 12·12-s − 4·13-s − 24·15-s − 2·16-s − 2·17-s + 30·18-s − 18·20-s + 8·23-s + 11·25-s − 12·26-s + 20·27-s + 14·29-s − 72·30-s − 12·31-s + 6·32-s − 6·34-s + 30·36-s − 6·37-s − 16·39-s + 6·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.30·3-s + 3/2·4-s − 2.68·5-s + 4.89·6-s + 10/3·9-s − 5.69·10-s + 3.46·12-s − 1.10·13-s − 6.19·15-s − 1/2·16-s − 0.485·17-s + 7.07·18-s − 4.02·20-s + 1.66·23-s + 11/5·25-s − 2.35·26-s + 3.84·27-s + 2.59·29-s − 13.1·30-s − 2.15·31-s + 1.06·32-s − 1.02·34-s + 5·36-s − 0.986·37-s − 2.56·39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(22.5818\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.026226455\)
\(L(\frac12)\) \(\approx\) \(6.026226455\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
good2$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} ) \)
5$C_2^2$ \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
29$C_2^2$ \( ( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 12 T + 115 T^{2} + 804 T^{3} + 5016 T^{4} + 804 p T^{5} + 115 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 6 T + 61 T^{2} + 294 T^{3} + 1476 T^{4} + 294 p T^{5} + 61 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 6 T + 69 T^{2} - 342 T^{3} + 2060 T^{4} - 342 p T^{5} + 69 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2^3$ \( 1 - 65 T^{2} + 2376 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 147 T^{2} - 1188 T^{3} + 9848 T^{4} - 1188 p T^{5} + 147 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 6 T + 5 T^{2} - 450 T^{3} - 3756 T^{4} - 450 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 24 T + 351 T^{2} + 3816 T^{3} + 33128 T^{4} + 3816 p T^{5} + 351 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 20 T^{2} + 3702 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 12 T + 139 T^{2} - 1092 T^{3} + 6648 T^{4} - 1092 p T^{5} + 139 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 12 T - 29 T^{2} - 180 T^{3} + 12312 T^{4} - 180 p T^{5} - 29 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 18 T + 301 T^{2} - 3474 T^{3} + 38316 T^{4} - 3474 p T^{5} + 301 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.774886395774201195497950970186, −8.212333687374287052917123440502, −8.031651163346810967388474299607, −7.84080404730479738269794113401, −7.75549703156030026352355146874, −7.61673933825580041068962048239, −7.02155404160861542453534409790, −6.87913316312645328137726413251, −6.77698578274062337517441281903, −6.44464104865362991986239792935, −5.93012579508360195881924611817, −5.42127565260545621576359315806, −5.04634307684214291959330798429, −4.92072686561926615260449599163, −4.68755313684334439612235489684, −4.32204017092603312116165338563, −4.15888800838187748054410391102, −3.88771264048786666034809925860, −3.62068152088635527120663925246, −3.36705152562684046556611750721, −2.93837800080755273840951834984, −2.89089078097043507743768710767, −2.30657838384625996078069108282, −1.83294260438404159532570875421, −0.76249376074704175307859539257, 0.76249376074704175307859539257, 1.83294260438404159532570875421, 2.30657838384625996078069108282, 2.89089078097043507743768710767, 2.93837800080755273840951834984, 3.36705152562684046556611750721, 3.62068152088635527120663925246, 3.88771264048786666034809925860, 4.15888800838187748054410391102, 4.32204017092603312116165338563, 4.68755313684334439612235489684, 4.92072686561926615260449599163, 5.04634307684214291959330798429, 5.42127565260545621576359315806, 5.93012579508360195881924611817, 6.44464104865362991986239792935, 6.77698578274062337517441281903, 6.87913316312645328137726413251, 7.02155404160861542453534409790, 7.61673933825580041068962048239, 7.75549703156030026352355146874, 7.84080404730479738269794113401, 8.031651163346810967388474299607, 8.212333687374287052917123440502, 8.774886395774201195497950970186

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