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Properties

Label 8-2e8-1.1-c12e4-0-0
Degree $8$
Conductor $256$
Sign $1$
Analytic cond. $178.654$
Root an. cond. $1.91206$
Motivic weight $12$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 108·2-s + 5.07e3·4-s − 1.83e4·5-s + 2.78e5·8-s + 3.80e5·9-s − 1.98e6·10-s − 7.32e6·13-s + 2.76e7·16-s + 5.92e7·17-s + 4.10e7·18-s − 9.31e7·20-s − 5.34e8·25-s − 7.90e8·26-s − 4.41e8·29-s + 1.57e9·32-s + 6.39e9·34-s + 1.92e9·36-s + 2.19e9·37-s − 5.10e9·40-s + 2.89e9·41-s − 6.97e9·45-s + 3.65e10·49-s − 5.77e10·50-s − 3.71e10·52-s − 1.49e10·53-s − 4.76e10·58-s − 8.37e10·61-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.23·4-s − 1.17·5-s + 1.06·8-s + 0.715·9-s − 1.98·10-s − 1.51·13-s + 1.64·16-s + 2.45·17-s + 1.20·18-s − 1.45·20-s − 2.18·25-s − 2.55·26-s − 0.741·29-s + 1.47·32-s + 4.14·34-s + 0.885·36-s + 0.856·37-s − 1.24·40-s + 0.609·41-s − 0.840·45-s + 2.63·49-s − 3.69·50-s − 1.87·52-s − 0.674·53-s − 1.25·58-s − 1.62·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(178.654\)
Root analytic conductor: \(1.91206\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 256,\ (\ :6, 6, 6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(4.824967961\)
\(L(\frac12)\) \(\approx\) \(4.824967961\)
\(L(7)\) 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$D_{4}$ \( 1 - 27 p^{2} T + 103 p^{6} T^{2} - 27 p^{14} T^{3} + p^{24} T^{4} \)
good3$C_2^2 \wr C_2$ \( 1 - 126700 p T^{2} - 25357898 p^{6} T^{4} - 126700 p^{25} T^{6} + p^{48} T^{8} \)
5$D_{4}$ \( ( 1 + 1836 p T + 3149918 p^{3} T^{2} + 1836 p^{13} T^{3} + p^{24} T^{4} )^{2} \)
7$C_2^2 \wr C_2$ \( 1 - 36519974020 T^{2} + 13500023147299253862 p^{2} T^{4} - 36519974020 p^{24} T^{6} + p^{48} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 - 25641130084 p^{2} T^{2} + \)\(89\!\cdots\!66\)\( p^{4} T^{4} - 25641130084 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 + 3660572 T + 48729122573862 T^{2} + 3660572 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 29623428 T + 1380562609561222 T^{2} - 29623428 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 - 3874871221824964 T^{2} + \)\(10\!\cdots\!66\)\( T^{4} - 3874871221824964 p^{24} T^{6} + p^{48} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 - 37255060909965700 T^{2} + \)\(22\!\cdots\!62\)\( p^{2} T^{4} - 37255060909965700 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 + 220510620 T + 433136148874755238 T^{2} + 220510620 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 - 1671958493177141764 T^{2} + \)\(13\!\cdots\!66\)\( T^{4} - 1671958493177141764 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 - 1099263268 T + 10446266562153682662 T^{2} - 1099263268 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 1448012484 T + 16485968390952224326 T^{2} - 1448012484 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!80\)\( T^{2} + \)\(69\!\cdots\!58\)\( T^{4} - \)\(12\!\cdots\!80\)\( p^{24} T^{6} + p^{48} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(19\!\cdots\!00\)\( T^{2} + \)\(30\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!00\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 + 7474220892 T + \)\(82\!\cdots\!82\)\( T^{2} + 7474220892 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(67\!\cdots\!04\)\( T^{2} + \)\(17\!\cdots\!26\)\( T^{4} - \)\(67\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 + 41882531996 T + \)\(38\!\cdots\!46\)\( T^{2} + 41882531996 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 - \)\(30\!\cdots\!20\)\( T^{2} + \)\(37\!\cdots\!18\)\( T^{4} - \)\(30\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(47\!\cdots\!84\)\( T^{2} + \)\(10\!\cdots\!26\)\( T^{4} - \)\(47\!\cdots\!84\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 - 402051989188 T + \)\(85\!\cdots\!42\)\( T^{2} - 402051989188 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + \)\(10\!\cdots\!16\)\( T^{2} + \)\(98\!\cdots\!26\)\( T^{4} + \)\(10\!\cdots\!16\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(28\!\cdots\!40\)\( T^{2} + \)\(39\!\cdots\!58\)\( T^{4} - \)\(28\!\cdots\!40\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 - 363502170180 T + \)\(48\!\cdots\!18\)\( T^{2} - 363502170180 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 1215890156548 T + \)\(15\!\cdots\!82\)\( T^{2} - 1215890156548 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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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

−16.59326636556613455665569238351, −15.74325440525839205848352435901, −15.35827818795853941977150770706, −15.09193550900033213370988424484, −14.56567982138910548959070610167, −13.91593105243877145839016333657, −13.89442184937612526468238388519, −13.10394757049963743784306396206, −12.43198164603991583016293699651, −12.21694767713942427770218362724, −12.00146758604148499364564888105, −11.21712170495027726050187740322, −10.50096852388341840474574081334, −9.800011674326689051692335821101, −9.510972574695669194449752822850, −7.896652903780465842880229192241, −7.67720139133811679677099022767, −7.40386975003873379608011877885, −6.03074551555212510479951314729, −5.37117840737053893438453537373, −4.66883199973894800937775704340, −3.84649909323247077742291957421, −3.54381533489840754041328877014, −2.12159529536355468119233187238, −0.72879330336756317193554392471, 0.72879330336756317193554392471, 2.12159529536355468119233187238, 3.54381533489840754041328877014, 3.84649909323247077742291957421, 4.66883199973894800937775704340, 5.37117840737053893438453537373, 6.03074551555212510479951314729, 7.40386975003873379608011877885, 7.67720139133811679677099022767, 7.896652903780465842880229192241, 9.510972574695669194449752822850, 9.800011674326689051692335821101, 10.50096852388341840474574081334, 11.21712170495027726050187740322, 12.00146758604148499364564888105, 12.21694767713942427770218362724, 12.43198164603991583016293699651, 13.10394757049963743784306396206, 13.89442184937612526468238388519, 13.91593105243877145839016333657, 14.56567982138910548959070610167, 15.09193550900033213370988424484, 15.35827818795853941977150770706, 15.74325440525839205848352435901, 16.59326636556613455665569238351

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