↑

Properties

Label 4-10e2-1.1-c27e2-0-0
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $2133.10$
Root an. cond. $6.79599$
Motivic weight $27$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.63e4·2-s − 2.24e6·3-s + 2.01e8·4-s − 2.44e9·5-s + 3.67e10·6-s − 1.20e11·7-s − 2.19e12·8-s − 4.81e12·9-s + 4.00e13·10-s + 7.11e12·11-s − 4.52e14·12-s − 3.23e14·13-s + 1.96e15·14-s + 5.48e15·15-s + 2.25e16·16-s − 1.19e16·17-s + 7.89e16·18-s − 3.10e17·19-s − 4.91e17·20-s + 2.69e17·21-s − 1.16e17·22-s − 1.20e18·23-s + 4.93e18·24-s + 4.47e18·25-s + 5.29e18·26-s + 1.58e19·27-s − 2.41e19·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.813·3-s + 3/2·4-s − 0.894·5-s + 1.15·6-s − 0.468·7-s − 1.41·8-s − 0.631·9-s + 1.26·10-s + 0.0621·11-s − 1.21·12-s − 0.295·13-s + 0.663·14-s + 0.727·15-s + 5/4·16-s − 0.292·17-s + 0.893·18-s − 1.69·19-s − 1.34·20-s + 0.381·21-s − 0.0878·22-s − 0.499·23-s + 1.15·24-s + 3/5·25-s + 0.418·26-s + 0.752·27-s − 0.703·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2133.10\)
Root analytic conductor: \(6.79599\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(0.4220808712\)
\(L(\frac12)\) \(\approx\) \(0.4220808712\)
\(L(\frac{29}{2})\) 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$C_1$ \( ( 1 + p^{13} T )^{2} \)
5$C_1$ \( ( 1 + p^{13} T )^{2} \)
good3$D_{4}$ \( 1 + 27724 p^{4} T + 1503044378 p^{8} T^{2} + 27724 p^{31} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 + 17170961756 p T + \)\(11\!\cdots\!98\)\( p^{2} T^{2} + 17170961756 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 - 58777873824 p^{2} T + \)\(21\!\cdots\!26\)\( p^{2} T^{2} - 58777873824 p^{29} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 24854999258108 p T + \)\(14\!\cdots\!02\)\( p^{2} T^{2} + 24854999258108 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 11949781720794252 T + \)\(16\!\cdots\!66\)\( p T^{2} + 11949781720794252 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 + 310800626187534200 T + \)\(35\!\cdots\!62\)\( p T^{2} + 310800626187534200 p^{27} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 1207992822578557644 T + \)\(51\!\cdots\!86\)\( p T^{2} + 1207992822578557644 p^{27} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 + 85089627076663724580 T + \)\(74\!\cdots\!18\)\( T^{2} + 85089627076663724580 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 + \)\(16\!\cdots\!56\)\( T + \)\(27\!\cdots\!06\)\( T^{2} + \)\(16\!\cdots\!56\)\( p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 - \)\(10\!\cdots\!08\)\( T + \)\(59\!\cdots\!82\)\( T^{2} - \)\(10\!\cdots\!08\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 - \)\(68\!\cdots\!44\)\( T + \)\(74\!\cdots\!46\)\( T^{2} - \)\(68\!\cdots\!44\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 - \)\(15\!\cdots\!36\)\( T + \)\(23\!\cdots\!38\)\( T^{2} - \)\(15\!\cdots\!36\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 - \)\(44\!\cdots\!28\)\( T + \)\(25\!\cdots\!22\)\( T^{2} - \)\(44\!\cdots\!28\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 - \)\(25\!\cdots\!96\)\( T + \)\(74\!\cdots\!78\)\( T^{2} - \)\(25\!\cdots\!96\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 - \)\(20\!\cdots\!40\)\( T + \)\(23\!\cdots\!38\)\( T^{2} - \)\(20\!\cdots\!40\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(67\!\cdots\!84\)\( T + \)\(28\!\cdots\!06\)\( T^{2} - \)\(67\!\cdots\!84\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 - \)\(24\!\cdots\!28\)\( T - \)\(59\!\cdots\!58\)\( T^{2} - \)\(24\!\cdots\!28\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 + \)\(23\!\cdots\!76\)\( T + \)\(33\!\cdots\!26\)\( T^{2} + \)\(23\!\cdots\!76\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 + \)\(13\!\cdots\!04\)\( T + \)\(20\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!04\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 + \)\(53\!\cdots\!20\)\( T + \)\(32\!\cdots\!18\)\( T^{2} + \)\(53\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 - \)\(57\!\cdots\!16\)\( T + \)\(31\!\cdots\!18\)\( T^{2} - \)\(57\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 - \)\(17\!\cdots\!20\)\( T + \)\(37\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 + \)\(21\!\cdots\!92\)\( T + \)\(62\!\cdots\!42\)\( T^{2} + \)\(21\!\cdots\!92\)\( p^{27} T^{3} + p^{54} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.81769808379749654003886499049, −14.65929143322800700159561592691, −13.00481352680701661230692550443, −12.53522425623574718283737721364, −11.52566781080487010264520674848, −11.32151222188806323740947560010, −10.62803714391003157110538564708, −9.899961410947716932522936102560, −8.811821741545700723240824981766, −8.676409457306965029706148662096, −7.46049827334676920740871767970, −7.19106737276169071822550298660, −5.98970753116883597123249698670, −5.79433923766414706649305681775, −4.36014088307428706865221189980, −3.62350515155519438926366452496, −2.51981501145865102016030578701, −1.96434064848990649254962466703, −0.57065644834970799583478072242, −0.44741907088208802246144618443, 0.44741907088208802246144618443, 0.57065644834970799583478072242, 1.96434064848990649254962466703, 2.51981501145865102016030578701, 3.62350515155519438926366452496, 4.36014088307428706865221189980, 5.79433923766414706649305681775, 5.98970753116883597123249698670, 7.19106737276169071822550298660, 7.46049827334676920740871767970, 8.676409457306965029706148662096, 8.811821741545700723240824981766, 9.899961410947716932522936102560, 10.62803714391003157110538564708, 11.32151222188806323740947560010, 11.52566781080487010264520674848, 12.53522425623574718283737721364, 13.00481352680701661230692550443, 14.65929143322800700159561592691, 14.81769808379749654003886499049

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