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Properties

Label 8-2645e4-1.1-c1e4-0-4
Degree $8$
Conductor $4.894\times 10^{13}$
Sign $1$
Analytic cond. $198980.$
Root an. cond. $4.59569$
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  − 2-s + 3-s − 3·4-s + 4·5-s − 6-s − 3·7-s + 2·8-s − 9-s − 4·10-s − 5·11-s − 3·12-s + 3·14-s + 4·15-s + 5·16-s + 5·17-s + 18-s − 4·19-s − 12·20-s − 3·21-s + 5·22-s + 2·24-s + 10·25-s − 7·27-s + 9·28-s − 5·29-s − 4·30-s − 13·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 3/2·4-s + 1.78·5-s − 0.408·6-s − 1.13·7-s + 0.707·8-s − 1/3·9-s − 1.26·10-s − 1.50·11-s − 0.866·12-s + 0.801·14-s + 1.03·15-s + 5/4·16-s + 1.21·17-s + 0.235·18-s − 0.917·19-s − 2.68·20-s − 0.654·21-s + 1.06·22-s + 0.408·24-s + 2·25-s − 1.34·27-s + 1.70·28-s − 0.928·29-s − 0.730·30-s − 2.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(198980.\)
Root analytic conductor: \(4.59569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{4} \cdot 23^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5$C_1$ \( ( 1 - T )^{4} \)
23 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + T + p^{2} T^{2} + 5 T^{3} + 5 p T^{4} + 5 p T^{5} + p^{4} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 - T + 2 T^{2} + 4 T^{3} - 8 T^{4} + 4 p T^{5} + 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 3 T + 8 T^{2} + 26 T^{3} + 120 T^{4} + 26 p T^{5} + 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 5 T + 49 T^{2} + 163 T^{3} + 835 T^{4} + 163 p T^{5} + 49 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 23 T^{2} + 75 T^{3} + 214 T^{4} + 75 p T^{5} + 23 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 5 T + 32 T^{2} - 90 T^{3} + 584 T^{4} - 90 p T^{5} + 32 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 4 T + 27 T^{2} + 137 T^{3} + 357 T^{4} + 137 p T^{5} + 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5 T + 53 T^{2} + 262 T^{3} + 2590 T^{4} + 262 p T^{5} + 53 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 13 T + 161 T^{2} + 1163 T^{3} + 7861 T^{4} + 1163 p T^{5} + 161 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 3 T + 110 T^{2} + 238 T^{3} + 5520 T^{4} + 238 p T^{5} + 110 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 20 T + 7 p T^{2} + 2739 T^{3} + 20345 T^{4} + 2739 p T^{5} + 7 p^{3} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 12 T + 205 T^{2} + 1529 T^{3} + 13844 T^{4} + 1529 p T^{5} + 205 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 9 T + 111 T^{2} + 524 T^{3} + 5012 T^{4} + 524 p T^{5} + 111 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 5 T + 118 T^{2} + 694 T^{3} + 7486 T^{4} + 694 p T^{5} + 118 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 4 T + 171 T^{2} + 791 T^{3} + 13404 T^{4} + 791 p T^{5} + 171 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 12 T + 278 T^{2} + 2176 T^{3} + 26415 T^{4} + 2176 p T^{5} + 278 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 18 T + 311 T^{2} + 2953 T^{3} + 444 p T^{4} + 2953 p T^{5} + 311 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 30 T + 585 T^{2} - 7567 T^{3} + 74555 T^{4} - 7567 p T^{5} + 585 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - T + 134 T^{2} - 420 T^{3} + 8898 T^{4} - 420 p T^{5} + 134 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 16 T + 278 T^{2} + 2984 T^{3} + 31759 T^{4} + 2984 p T^{5} + 278 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 24 T + 449 T^{2} + 5977 T^{3} + 60332 T^{4} + 5977 p T^{5} + 449 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 9 T + 266 T^{2} - 2012 T^{3} + 32324 T^{4} - 2012 p T^{5} + 266 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 39 T + 898 T^{2} + 13972 T^{3} + 159698 T^{4} + 13972 p T^{5} + 898 p^{2} T^{6} + 39 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.50769892645790461776068840478, −6.49012891397422744388386363300, −6.43607975420617370550743697198, −6.02883702912453338451524249643, −5.78364561279797780483892218219, −5.45210748117731989329463733492, −5.41432621591596608609811342149, −5.36986743278856273267494725393, −5.33487503871263212221786580598, −4.78385116711603821090488473087, −4.75291129089752506757958696255, −4.61212711165320839833457613075, −3.95932078865498064571508981630, −3.78911273505231033535901354311, −3.68985934851263286669162306491, −3.67000917380202721269223696867, −3.09688780358634569674781459335, −2.99157631501289813720128463816, −2.80893337186891956309651335218, −2.60094031207873859677552328914, −2.34016502694041441072807132808, −1.72888800119141259946725153568, −1.63231749224219710093304436141, −1.53322279139139565312813755483, −1.25689706438514459069855289791, 0, 0, 0, 0, 1.25689706438514459069855289791, 1.53322279139139565312813755483, 1.63231749224219710093304436141, 1.72888800119141259946725153568, 2.34016502694041441072807132808, 2.60094031207873859677552328914, 2.80893337186891956309651335218, 2.99157631501289813720128463816, 3.09688780358634569674781459335, 3.67000917380202721269223696867, 3.68985934851263286669162306491, 3.78911273505231033535901354311, 3.95932078865498064571508981630, 4.61212711165320839833457613075, 4.75291129089752506757958696255, 4.78385116711603821090488473087, 5.33487503871263212221786580598, 5.36986743278856273267494725393, 5.41432621591596608609811342149, 5.45210748117731989329463733492, 5.78364561279797780483892218219, 6.02883702912453338451524249643, 6.43607975420617370550743697198, 6.49012891397422744388386363300, 6.50769892645790461776068840478

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