Properties

Label 8-96e8-1.1-c1e4-0-14
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 4·7-s + 8·13-s − 12·29-s − 12·31-s + 16·35-s + 16·37-s − 4·49-s − 20·53-s + 16·61-s − 32·65-s + 16·67-s − 8·71-s − 8·73-s − 12·79-s + 8·89-s − 32·91-s − 20·101-s − 4·103-s + 24·109-s + 8·113-s − 20·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.51·7-s + 2.21·13-s − 2.22·29-s − 2.15·31-s + 2.70·35-s + 2.63·37-s − 4/7·49-s − 2.74·53-s + 2.04·61-s − 3.96·65-s + 1.95·67-s − 0.949·71-s − 0.936·73-s − 1.35·79-s + 0.847·89-s − 3.35·91-s − 1.99·101-s − 0.394·103-s + 2.29·109-s + 0.752·113-s − 1.81·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
3 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 16 T^{2} + 44 T^{3} + 118 T^{4} + 44 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} + 60 T^{3} + 186 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} + 32 T^{3} + 230 T^{4} + 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 56 T^{2} - 264 T^{3} + 1122 T^{4} - 264 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 44 T^{2} - 64 T^{3} + 966 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 144 T^{2} + 964 T^{3} + 6422 T^{4} + 964 p T^{5} + 144 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 164 T^{2} + 1140 T^{3} + 8218 T^{4} + 1140 p T^{5} + 164 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 200 T^{2} - 1552 T^{3} + 11010 T^{4} - 1552 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 100 T^{2} - 192 T^{3} + 4726 T^{4} - 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 76 T^{2} - 256 T^{3} + 2726 T^{4} - 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 336 T^{2} + 3452 T^{3} + 30134 T^{4} + 3452 p T^{5} + 336 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 296 T^{2} - 2704 T^{3} + 27618 T^{4} - 2704 p T^{5} + 296 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 4 p T^{2} - 2960 T^{3} + 27190 T^{4} - 2960 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 252 T^{2} + 1512 T^{3} + 25766 T^{4} + 1512 p T^{5} + 252 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 44 p T^{5} + 148 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 116 T^{2} + 160 T^{3} + 5510 T^{4} + 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 156 T^{2} - 504 T^{3} + 10022 T^{4} - 504 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.85044721164231395663591154929, −5.63514148948046398475165084109, −5.56752196075931279773074056946, −5.18989499461168515496968654952, −5.16964524345623687718873763114, −4.66644228329550842986934866634, −4.58610693823984631037182861285, −4.40499414030397629699523691258, −4.37862506242163473953890770591, −3.80622012485845479039353756666, −3.79438763967567244417278931813, −3.77703259601728898003175729745, −3.76331679385374910476024192803, −3.28068033205754038989498029927, −3.27715767496555758821559394971, −3.14157221243803111160534895048, −3.01989908625730467010469919970, −2.34244573133933266322068200011, −2.27534018618167788497511261890, −2.11611777592116577652830938193, −2.08316580306559325955819193360, −1.39286392348086932546610330852, −1.22553815751841969546129116006, −1.15913802002400196484589656062, −0.936160522049080287652222341842, 0, 0, 0, 0, 0.936160522049080287652222341842, 1.15913802002400196484589656062, 1.22553815751841969546129116006, 1.39286392348086932546610330852, 2.08316580306559325955819193360, 2.11611777592116577652830938193, 2.27534018618167788497511261890, 2.34244573133933266322068200011, 3.01989908625730467010469919970, 3.14157221243803111160534895048, 3.27715767496555758821559394971, 3.28068033205754038989498029927, 3.76331679385374910476024192803, 3.77703259601728898003175729745, 3.79438763967567244417278931813, 3.80622012485845479039353756666, 4.37862506242163473953890770591, 4.40499414030397629699523691258, 4.58610693823984631037182861285, 4.66644228329550842986934866634, 5.16964524345623687718873763114, 5.18989499461168515496968654952, 5.56752196075931279773074056946, 5.63514148948046398475165084109, 5.85044721164231395663591154929

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