↑

Properties

Label 8-483e4-1.1-c3e4-0-0
Degree $8$
Conductor $54423757521$
Sign $1$
Analytic cond. $659557.$
Root an. cond. $5.33834$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 12·3-s − 9·4-s − 23·5-s + 24·6-s − 28·7-s − 16·8-s + 90·9-s − 46·10-s − 48·11-s − 108·12-s − 107·13-s − 56·14-s − 276·15-s − 39·16-s − 6·17-s + 180·18-s + 76·19-s + 207·20-s − 336·21-s − 96·22-s − 92·23-s − 192·24-s + 72·25-s − 214·26-s + 540·27-s + 252·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s − 9/8·4-s − 2.05·5-s + 1.63·6-s − 1.51·7-s − 0.707·8-s + 10/3·9-s − 1.45·10-s − 1.31·11-s − 2.59·12-s − 2.28·13-s − 1.06·14-s − 4.75·15-s − 0.609·16-s − 0.0856·17-s + 2.35·18-s + 0.917·19-s + 2.31·20-s − 3.49·21-s − 0.930·22-s − 0.834·23-s − 1.63·24-s + 0.575·25-s − 1.61·26-s + 3.84·27-s + 1.70·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(659557.\)
Root analytic conductor: \(5.33834\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
23$C_1$ \( ( 1 + p T )^{4} \)
good2$D_{4}$ \( ( 1 - T + 3 p T^{2} - p^{3} T^{3} + p^{6} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 23 T + 457 T^{2} + 5887 T^{3} + 72576 T^{4} + 5887 p^{3} T^{5} + 457 p^{6} T^{6} + 23 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 48 T + 5821 T^{2} + 190506 T^{3} + 11938596 T^{4} + 190506 p^{3} T^{5} + 5821 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 107 T + 6309 T^{2} + 184627 T^{3} + 5842328 T^{4} + 184627 p^{3} T^{5} + 6309 p^{6} T^{6} + 107 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 8099 T^{2} + 261864 T^{3} + 33593436 T^{4} + 261864 p^{3} T^{5} + 8099 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 4 p T + 3387 T^{2} + 208072 T^{3} + 10709276 T^{4} + 208072 p^{3} T^{5} + 3387 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 204 T + 89947 T^{2} - 11764032 T^{3} + 3060953256 T^{4} - 11764032 p^{3} T^{5} + 89947 p^{6} T^{6} - 204 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 276 T + 104785 T^{2} + 24332022 T^{3} + 4481356764 T^{4} + 24332022 p^{3} T^{5} + 104785 p^{6} T^{6} + 276 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 248 T + 68833 T^{2} - 7264474 T^{3} - 1502500976 T^{4} - 7264474 p^{3} T^{5} + 68833 p^{6} T^{6} + 248 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 450 T + 287281 T^{2} + 88794354 T^{3} + 30193477812 T^{4} + 88794354 p^{3} T^{5} + 287281 p^{6} T^{6} + 450 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 109 T + 178431 T^{2} - 5659427 T^{3} + 17230772360 T^{4} - 5659427 p^{3} T^{5} + 178431 p^{6} T^{6} - 109 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 688 T + 577780 T^{2} + 229100120 T^{3} + 98785113750 T^{4} + 229100120 p^{3} T^{5} + 577780 p^{6} T^{6} + 688 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 633 T + 668407 T^{2} + 281904165 T^{3} + 154511433492 T^{4} + 281904165 p^{3} T^{5} + 668407 p^{6} T^{6} + 633 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 585 T + 838471 T^{2} - 357781101 T^{3} + 259978127064 T^{4} - 357781101 p^{3} T^{5} + 838471 p^{6} T^{6} - 585 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 1311 T + 1318645 T^{2} - 837877239 T^{3} + 466242910200 T^{4} - 837877239 p^{3} T^{5} + 1318645 p^{6} T^{6} - 1311 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 365 T + 226797 T^{2} - 32351755 T^{3} - 59441665780 T^{4} - 32351755 p^{3} T^{5} + 226797 p^{6} T^{6} - 365 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 3049 T + 4776115 T^{2} + 4820961905 T^{3} + 3413277866856 T^{4} + 4820961905 p^{3} T^{5} + 4776115 p^{6} T^{6} + 3049 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 1164 T + 463243 T^{2} - 200075904 T^{3} - 275333190672 T^{4} - 200075904 p^{3} T^{5} + 463243 p^{6} T^{6} + 1164 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 476 T + 1594059 T^{2} + 624085564 T^{3} + 1101135024008 T^{4} + 624085564 p^{3} T^{5} + 1594059 p^{6} T^{6} + 476 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 1462 T + 2341009 T^{2} - 1833929162 T^{3} + 1786336041912 T^{4} - 1833929162 p^{3} T^{5} + 2341009 p^{6} T^{6} - 1462 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1767 T + 2518495 T^{2} + 2026105023 T^{3} + 1914293102940 T^{4} + 2026105023 p^{3} T^{5} + 2518495 p^{6} T^{6} + 1767 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 1788 T + 3523081 T^{2} - 3930682914 T^{3} + 4744768098480 T^{4} - 3930682914 p^{3} T^{5} + 3523081 p^{6} T^{6} - 1788 p^{9} T^{7} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.148408946972322093130377675807, −7.65523948166021950572783036533, −7.54647051998256044307912707361, −7.29937557427566638465494717008, −7.25461993774901886206589840520, −7.01010355813711154498826316161, −6.72489474754728443614729932487, −6.25749778853553382206832395794, −6.22612499891496431833892294750, −5.36323110422934097881086760294, −5.23274762230016573436888978801, −5.13738593728931244162745467111, −5.01622764424536686654508928635, −4.37666341569694277220701977527, −4.30157793903978839311626841540, −4.08743355523628724410779249690, −3.90506151367073209296134336369, −3.45749051856508233629037826962, −3.32908642876575004106064498578, −3.16420591363827880420440490887, −2.68143401051434301243510148679, −2.42196336507189864477555674313, −2.29886995031709797061655751692, −1.65343048168390364757529718047, −1.24932549400161210994993597944, 0, 0, 0, 0, 1.24932549400161210994993597944, 1.65343048168390364757529718047, 2.29886995031709797061655751692, 2.42196336507189864477555674313, 2.68143401051434301243510148679, 3.16420591363827880420440490887, 3.32908642876575004106064498578, 3.45749051856508233629037826962, 3.90506151367073209296134336369, 4.08743355523628724410779249690, 4.30157793903978839311626841540, 4.37666341569694277220701977527, 5.01622764424536686654508928635, 5.13738593728931244162745467111, 5.23274762230016573436888978801, 5.36323110422934097881086760294, 6.22612499891496431833892294750, 6.25749778853553382206832395794, 6.72489474754728443614729932487, 7.01010355813711154498826316161, 7.25461993774901886206589840520, 7.29937557427566638465494717008, 7.54647051998256044307912707361, 7.65523948166021950572783036533, 8.148408946972322093130377675807

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