Properties

Label 8-1008e4-1.1-c1e4-0-11
Degree $8$
Conductor $1.032\times 10^{12}$
Sign $1$
Analytic cond. $4197.11$
Root an. cond. $2.83706$
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  + 3-s − 3·5-s + 2·7-s + 3·9-s + 3·11-s − 4·13-s − 3·15-s − 6·17-s − 20·19-s + 2·21-s − 9·23-s + 4·25-s + 8·27-s + 6·29-s + 4·31-s + 3·33-s − 6·35-s + 8·37-s − 4·39-s − 15·41-s + 43-s − 9·45-s + 49-s − 6·51-s − 12·53-s − 9·55-s − 20·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 0.755·7-s + 9-s + 0.904·11-s − 1.10·13-s − 0.774·15-s − 1.45·17-s − 4.58·19-s + 0.436·21-s − 1.87·23-s + 4/5·25-s + 1.53·27-s + 1.11·29-s + 0.718·31-s + 0.522·33-s − 1.01·35-s + 1.31·37-s − 0.640·39-s − 2.34·41-s + 0.152·43-s − 1.34·45-s + 1/7·49-s − 0.840·51-s − 1.64·53-s − 1.21·55-s − 2.64·57-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

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4197.11\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.041240228\)
\(L(\frac12)\) \(\approx\) \(2.041240228\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good5$C_2$$\times$$C_2^2$ \( ( 1 + 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
17$D_{4}$ \( ( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 9 T + p T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} ) \)
29$D_4\times C_2$ \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 + 15 T + 95 T^{2} + 720 T^{3} + 5994 T^{4} + 720 p T^{5} + 95 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - T - 11 T^{2} + 74 T^{3} - 1748 T^{4} + 74 p T^{5} - 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 216 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 484 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \)
71$D_{4}$ \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 7 T - 47 T^{2} + 434 T^{3} - 896 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.01332856906342144192493938645, −6.92893658300723183552476015433, −6.66660838196899357486294182498, −6.39057440179189247757570021477, −6.35319007362709404866567951414, −6.35131370279724072275923104060, −5.96091345001901944815310780112, −5.57064046654025726029633112489, −4.90028721063857012513000524765, −4.86468183168733485620411339317, −4.77792939462161263347259449504, −4.44484388581840663847271241772, −4.41351878975251815500381550245, −4.23128876236959883134571761448, −3.76144549455249823500204146326, −3.70517479695938801032386557591, −3.41545300107938504884513458680, −3.06037505566931324819915046008, −2.39720366973919982559689031396, −2.26307694191964373997342136842, −2.15935449074153625626220979903, −1.93704145215016857769358072235, −1.52446479141954707100979751003, −0.63088950456251519190276872361, −0.44980496150069241603862583430, 0.44980496150069241603862583430, 0.63088950456251519190276872361, 1.52446479141954707100979751003, 1.93704145215016857769358072235, 2.15935449074153625626220979903, 2.26307694191964373997342136842, 2.39720366973919982559689031396, 3.06037505566931324819915046008, 3.41545300107938504884513458680, 3.70517479695938801032386557591, 3.76144549455249823500204146326, 4.23128876236959883134571761448, 4.41351878975251815500381550245, 4.44484388581840663847271241772, 4.77792939462161263347259449504, 4.86468183168733485620411339317, 4.90028721063857012513000524765, 5.57064046654025726029633112489, 5.96091345001901944815310780112, 6.35131370279724072275923104060, 6.35319007362709404866567951414, 6.39057440179189247757570021477, 6.66660838196899357486294182498, 6.92893658300723183552476015433, 7.01332856906342144192493938645

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