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Properties

Label 8-3750e4-1.1-c1e4-0-15
Degree $8$
Conductor $1.978\times 10^{14}$
Sign $1$
Analytic cond. $803958.$
Root an. cond. $5.47210$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·3-s + 10·4-s + 16·6-s − 8·7-s + 20·8-s + 10·9-s − 12·11-s + 40·12-s − 14·13-s − 32·14-s + 35·16-s − 3·17-s + 40·18-s − 5·19-s − 32·21-s − 48·22-s − 9·23-s + 80·24-s − 56·26-s + 20·27-s − 80·28-s − 2·31-s + 56·32-s − 48·33-s − 12·34-s + 100·36-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 6.53·6-s − 3.02·7-s + 7.07·8-s + 10/3·9-s − 3.61·11-s + 11.5·12-s − 3.88·13-s − 8.55·14-s + 35/4·16-s − 0.727·17-s + 9.42·18-s − 1.14·19-s − 6.98·21-s − 10.2·22-s − 1.87·23-s + 16.3·24-s − 10.9·26-s + 3.84·27-s − 15.1·28-s − 0.359·31-s + 9.89·32-s − 8.35·33-s − 2.05·34-s + 50/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(803958.\)
Root analytic conductor: \(5.47210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
good7$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 6 p T^{2} + 145 T^{3} + 431 T^{4} + 145 p T^{5} + 6 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 83 T^{2} + 414 T^{3} + 1575 T^{4} + 414 p T^{5} + 83 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 108 T^{2} + 580 T^{3} + 2381 T^{4} + 580 p T^{5} + 108 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + p T^{2} - 90 T^{3} - 279 T^{4} - 90 p T^{5} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 36 T^{2} + 85 T^{3} + 491 T^{4} + 85 p T^{5} + 36 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 98 T^{2} + 495 T^{3} + 3171 T^{4} + 495 p T^{5} + 98 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 + 86 T^{2} + 3351 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 3 p T^{2} + 214 T^{3} + 125 p T^{4} + 214 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 267 T^{2} + 2110 T^{3} + 13721 T^{4} + 2110 p T^{5} + 267 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 143 T^{2} + 1134 T^{3} + 7815 T^{4} + 1134 p T^{5} + 143 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 138 T^{2} + 895 T^{3} + 6401 T^{4} + 895 p T^{5} + 138 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 182 T^{2} + 405 T^{3} + 12681 T^{4} + 405 p T^{5} + 182 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 158 T^{2} + 945 T^{3} + 10401 T^{4} + 945 p T^{5} + 158 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 15 T + 221 T^{2} + 1980 T^{3} + 18891 T^{4} + 1980 p T^{5} + 221 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 108 T^{2} - 1241 T^{3} + 13535 T^{4} - 1241 p T^{5} + 108 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 447 T^{2} + 5170 T^{3} + 51461 T^{4} + 5170 p T^{5} + 447 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 218 T^{2} + 1989 T^{3} + 21405 T^{4} + 1989 p T^{5} + 218 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 228 T^{2} + 2335 T^{3} + 22121 T^{4} + 2335 p T^{5} + 228 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 306 T^{2} + 1135 T^{3} + 35861 T^{4} + 1135 p T^{5} + 306 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 38 T^{2} + 405 T^{3} - 6939 T^{4} + 405 p T^{5} + 38 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 15 T + 296 T^{2} + 2205 T^{3} + 29871 T^{4} + 2205 p T^{5} + 296 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 252 T^{2} + 2545 T^{3} + 28031 T^{4} + 2545 p T^{5} + 252 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.65650506417384023849324217520, −6.08160923811809271032061513193, −5.94629847429992111483265960100, −5.71462310814665519468684628103, −5.53501504810168258848376470629, −5.31538011865217244464916650016, −5.11724684028908337699031853314, −4.89829668960837727485278347957, −4.89774889650597722399466553051, −4.42478444061377337374175635308, −4.40800702286941243653781045551, −4.31275061295516396820541522297, −4.07182209735679580362672695534, −3.42124526536376324571087963516, −3.33209219330913395309863055428, −3.30851711398363703572460017173, −3.23554362814402574788384995478, −2.86426326433242728059825928330, −2.79772144685195123423355367183, −2.79040675306761924857703386374, −2.32856781189070118848217097682, −2.11501287284011715408969642675, −1.82259157794265517084524842757, −1.80281890027780455481067128756, −1.76872180241912204418222385642, 0, 0, 0, 0, 1.76872180241912204418222385642, 1.80281890027780455481067128756, 1.82259157794265517084524842757, 2.11501287284011715408969642675, 2.32856781189070118848217097682, 2.79040675306761924857703386374, 2.79772144685195123423355367183, 2.86426326433242728059825928330, 3.23554362814402574788384995478, 3.30851711398363703572460017173, 3.33209219330913395309863055428, 3.42124526536376324571087963516, 4.07182209735679580362672695534, 4.31275061295516396820541522297, 4.40800702286941243653781045551, 4.42478444061377337374175635308, 4.89774889650597722399466553051, 4.89829668960837727485278347957, 5.11724684028908337699031853314, 5.31538011865217244464916650016, 5.53501504810168258848376470629, 5.71462310814665519468684628103, 5.94629847429992111483265960100, 6.08160923811809271032061513193, 6.65650506417384023849324217520

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