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Properties

Label 4-2e8-1.1-c25e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $4014.42$
Root an. cond. $7.95986$
Motivic weight $25$
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  − 3.79e5·3-s + 7.41e8·5-s + 3.76e8·7-s + 8.72e11·9-s − 8.32e12·11-s − 1.06e14·13-s − 2.81e14·15-s + 1.32e15·17-s + 4.77e14·19-s − 1.43e14·21-s + 1.15e17·23-s + 7.49e16·25-s − 9.29e17·27-s + 1.72e18·29-s + 8.68e18·31-s + 3.16e18·33-s + 2.79e17·35-s − 3.49e19·37-s + 4.04e19·39-s + 8.30e19·41-s − 4.45e19·43-s + 6.47e20·45-s − 1.86e21·47-s − 1.76e21·49-s − 5.04e20·51-s − 3.24e21·53-s − 6.17e21·55-s + ⋯
L(s)  = 1  − 0.412·3-s + 1.35·5-s + 0.0102·7-s + 1.02·9-s − 0.799·11-s − 1.26·13-s − 0.560·15-s + 0.552·17-s + 0.0494·19-s − 0.00424·21-s + 1.09·23-s + 0.251·25-s − 1.19·27-s + 0.905·29-s + 1.98·31-s + 0.329·33-s + 0.0139·35-s − 0.871·37-s + 0.523·39-s + 0.574·41-s − 0.169·43-s + 1.39·45-s − 2.34·47-s − 1.31·49-s − 0.228·51-s − 0.906·53-s − 1.08·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(4014.42\)
Root analytic conductor: \(7.95986\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(2.897092258\)
\(L(\frac12)\) \(\approx\) \(2.897092258\)
\(L(\frac{27}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 + 126616 p T - 332800574 p^{7} T^{2} + 126616 p^{26} T^{3} + p^{50} T^{4} \)
5$D_{4}$ \( 1 - 29678124 p^{2} T + 760941305600974 p^{4} T^{2} - 29678124 p^{27} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 - 53790992 p T + 736567609831341198 p^{4} T^{2} - 53790992 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 + 756639510024 p T + \)\(17\!\cdots\!46\)\( p^{3} T^{2} + 756639510024 p^{26} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 106467053152292 T + \)\(11\!\cdots\!54\)\( p T^{2} + 106467053152292 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 - 1327878920113956 T + \)\(39\!\cdots\!94\)\( p T^{2} - 1327878920113956 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 - 477079242949400 T + \)\(83\!\cdots\!42\)\( p T^{2} - 477079242949400 p^{25} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 - 5013261990498864 p T + \)\(43\!\cdots\!58\)\( p^{2} T^{2} - 5013261990498864 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 - 1724412645206435580 T + \)\(30\!\cdots\!98\)\( T^{2} - 1724412645206435580 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 - 8688082288351126976 T + \)\(57\!\cdots\!46\)\( T^{2} - 8688082288351126976 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 + 34908364049750170484 T + \)\(27\!\cdots\!78\)\( T^{2} + 34908364049750170484 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 - 83014324355953468884 T + \)\(29\!\cdots\!66\)\( T^{2} - 83014324355953468884 p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 + 44539608583471901848 T + \)\(11\!\cdots\!62\)\( T^{2} + 44539608583471901848 p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 + 39773632596907970208 p T + \)\(19\!\cdots\!58\)\( T^{2} + 39773632596907970208 p^{26} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 + \)\(32\!\cdots\!32\)\( T + \)\(28\!\cdots\!42\)\( T^{2} + \)\(32\!\cdots\!32\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(42\!\cdots\!98\)\( T^{2} - \)\(17\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 - \)\(33\!\cdots\!44\)\( T + \)\(85\!\cdots\!86\)\( T^{2} - \)\(33\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 + \)\(33\!\cdots\!16\)\( T + \)\(43\!\cdots\!78\)\( T^{2} + \)\(33\!\cdots\!16\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 - \)\(27\!\cdots\!56\)\( T + \)\(49\!\cdots\!86\)\( T^{2} - \)\(27\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(31\!\cdots\!48\)\( T + \)\(86\!\cdots\!62\)\( T^{2} - \)\(31\!\cdots\!48\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 + \)\(92\!\cdots\!20\)\( T + \)\(73\!\cdots\!98\)\( T^{2} + \)\(92\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 - \)\(45\!\cdots\!52\)\( T - \)\(76\!\cdots\!38\)\( T^{2} - \)\(45\!\cdots\!52\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 + \)\(23\!\cdots\!20\)\( T + \)\(80\!\cdots\!98\)\( T^{2} + \)\(23\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 - \)\(13\!\cdots\!36\)\( T + \)\(12\!\cdots\!38\)\( T^{2} - \)\(13\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.67207751895499161358803043245, −13.00108795312898002748002726416, −12.68110897464289691074530459316, −11.83373950796997461601512574218, −11.10728760474328901753007745851, −10.12032266875077355438789953625, −9.906756360376971332201207381686, −9.561469433521880245116809333716, −8.320944878457620311042498748738, −7.72919487931930093605223924110, −6.75131635763864416480189744610, −6.47674008798368008279193010116, −5.28385380013218117670821177625, −5.18097272429796039104452829655, −4.37794473092975835173861452757, −3.21234176240876412265908682906, −2.52544268716344207757714854446, −1.87338149733849407912840329629, −1.21174484783732736854979448042, −0.44782006828610465679020928063, 0.44782006828610465679020928063, 1.21174484783732736854979448042, 1.87338149733849407912840329629, 2.52544268716344207757714854446, 3.21234176240876412265908682906, 4.37794473092975835173861452757, 5.18097272429796039104452829655, 5.28385380013218117670821177625, 6.47674008798368008279193010116, 6.75131635763864416480189744610, 7.72919487931930093605223924110, 8.320944878457620311042498748738, 9.561469433521880245116809333716, 9.906756360376971332201207381686, 10.12032266875077355438789953625, 11.10728760474328901753007745851, 11.83373950796997461601512574218, 12.68110897464289691074530459316, 13.00108795312898002748002726416, 13.67207751895499161358803043245

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