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Properties

Label 8-363e4-1.1-c2e4-0-1
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $9571.20$
Root an. cond. $3.14500$
Motivic weight $2$
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  + 3·3-s − 4·4-s + 11·7-s − 12·12-s − 22·13-s + 11·19-s + 33·21-s − 25·25-s − 44·28-s − 59·31-s − 47·37-s − 66·39-s + 88·43-s + 49·49-s + 88·52-s + 33·57-s − 121·61-s − 52·67-s + 143·73-s − 75·75-s − 44·76-s + 11·79-s − 132·84-s − 242·91-s − 177·93-s + 169·97-s + 100·100-s + ⋯
L(s)  = 1  + 3-s − 4-s + 11/7·7-s − 12-s − 1.69·13-s + 0.578·19-s + 11/7·21-s − 25-s − 1.57·28-s − 1.90·31-s − 1.27·37-s − 1.69·39-s + 2.04·43-s + 49-s + 1.69·52-s + 0.578·57-s − 1.98·61-s − 0.776·67-s + 1.95·73-s − 75-s − 0.578·76-s + 0.139·79-s − 1.57·84-s − 2.65·91-s − 1.90·93-s + 1.74·97-s + 100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(9571.20\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4814974928\)
\(L(\frac12)\) \(\approx\) \(0.4814974928\)
\(L(2)\) 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$C_4$ \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} \)
11 \( 1 \)
good2$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
5$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
7$C_4\times C_2$ \( 1 - 11 T + 72 T^{2} - 253 T^{3} - 745 T^{4} - 253 p^{2} T^{5} + 72 p^{4} T^{6} - 11 p^{6} T^{7} + p^{8} T^{8} \)
13$C_4\times C_2$ \( 1 + 22 T + 315 T^{2} + 3212 T^{3} + 17429 T^{4} + 3212 p^{2} T^{5} + 315 p^{4} T^{6} + 22 p^{6} T^{7} + p^{8} T^{8} \)
17$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
19$C_4\times C_2$ \( 1 - 11 T - 240 T^{2} + 6611 T^{3} + 13919 T^{4} + 6611 p^{2} T^{5} - 240 p^{4} T^{6} - 11 p^{6} T^{7} + p^{8} T^{8} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
31$C_4\times C_2$ \( 1 + 59 T + 2520 T^{2} + 91981 T^{3} + 3005159 T^{4} + 91981 p^{2} T^{5} + 2520 p^{4} T^{6} + 59 p^{6} T^{7} + p^{8} T^{8} \)
37$C_4\times C_2$ \( 1 + 47 T + 840 T^{2} - 24863 T^{3} - 2318521 T^{4} - 24863 p^{2} T^{5} + 840 p^{4} T^{6} + 47 p^{6} T^{7} + p^{8} T^{8} \)
41$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
43$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
61$C_4\times C_2$ \( 1 + 121 T + 10920 T^{2} + 871079 T^{3} + 64767239 T^{4} + 871079 p^{2} T^{5} + 10920 p^{4} T^{6} + 121 p^{6} T^{7} + p^{8} T^{8} \)
67$C_2$ \( ( 1 + 13 T + p^{2} T^{2} )^{4} \)
71$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
73$C_4\times C_2$ \( 1 - 143 T + 15120 T^{2} - 1400113 T^{3} + 119641679 T^{4} - 1400113 p^{2} T^{5} + 15120 p^{4} T^{6} - 143 p^{6} T^{7} + p^{8} T^{8} \)
79$C_4\times C_2$ \( 1 - 11 T - 6120 T^{2} + 135971 T^{3} + 36699239 T^{4} + 135971 p^{2} T^{5} - 6120 p^{4} T^{6} - 11 p^{6} T^{7} + p^{8} T^{8} \)
83$C_4$$\times$$C_4$ \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
97$C_4\times C_2$ \( 1 - 169 T + 19152 T^{2} - 1646567 T^{3} + 98068655 T^{4} - 1646567 p^{2} T^{5} + 19152 p^{4} T^{6} - 169 p^{6} T^{7} + p^{8} T^{8} \)
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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

−7.977995075790014923488877862851, −7.970306569573404670128460428574, −7.62535905706536169615552341177, −7.38283166378846022832708663146, −7.29241032243578393937197267644, −6.95716202983744500082611415111, −6.67341574700158000740152775873, −6.14138610955154147233063594157, −5.93066785938027438798963794168, −5.70503123515228906029902442337, −5.25571288478607986485766226376, −5.01936012241379553689404158606, −4.98537508723023379003924166893, −4.68895959613241906278253609728, −4.33016000537224231018635651305, −3.85854626850095864416095372911, −3.84462764381206909934726529027, −3.48638145136014149434363084150, −2.85563519995309457242950845716, −2.67174761556297139425785480829, −2.35458388331983318426914553385, −1.81344253694140428016764957541, −1.66593026482930875761614946093, −0.999783355947410086190317290308, −0.14388886121397486654340817929, 0.14388886121397486654340817929, 0.999783355947410086190317290308, 1.66593026482930875761614946093, 1.81344253694140428016764957541, 2.35458388331983318426914553385, 2.67174761556297139425785480829, 2.85563519995309457242950845716, 3.48638145136014149434363084150, 3.84462764381206909934726529027, 3.85854626850095864416095372911, 4.33016000537224231018635651305, 4.68895959613241906278253609728, 4.98537508723023379003924166893, 5.01936012241379553689404158606, 5.25571288478607986485766226376, 5.70503123515228906029902442337, 5.93066785938027438798963794168, 6.14138610955154147233063594157, 6.67341574700158000740152775873, 6.95716202983744500082611415111, 7.29241032243578393937197267644, 7.38283166378846022832708663146, 7.62535905706536169615552341177, 7.970306569573404670128460428574, 7.977995075790014923488877862851

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