Properties

Degree 10
Conductor $ 5^{5} $
Sign $-1$
Motivic weight 33
Primitive no
Self-dual yes
Analytic rank 5

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.04e4·2-s − 1.49e7·3-s − 2.04e10·4-s − 7.62e11·5-s − 4.56e11·6-s − 6.54e13·7-s − 6.67e14·8-s − 6.51e15·9-s − 2.32e16·10-s − 2.87e17·11-s + 3.06e17·12-s + 2.39e18·13-s − 1.99e18·14-s + 1.14e19·15-s + 1.82e20·16-s − 3.08e20·17-s − 1.98e20·18-s + 9.18e20·19-s + 1.55e22·20-s + 9.81e20·21-s − 8.76e21·22-s + 5.54e22·23-s + 1.00e22·24-s + 3.49e23·25-s + 7.30e22·26-s + 3.52e23·27-s + 1.33e24·28-s + ⋯
L(s)  = 1  + 0.328·2-s − 0.201·3-s − 2.37·4-s − 2.23·5-s − 0.0660·6-s − 0.744·7-s − 0.838·8-s − 1.17·9-s − 0.735·10-s − 1.88·11-s + 0.478·12-s + 0.999·13-s − 0.244·14-s + 0.449·15-s + 2.48·16-s − 1.53·17-s − 0.385·18-s + 0.730·19-s + 5.32·20-s + 0.149·21-s − 0.620·22-s + 1.88·23-s + 0.168·24-s + 3·25-s + 0.328·26-s + 0.851·27-s + 1.77·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+33/2)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(3125\)    =    \(5^{5}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(33\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  5
Selberg data  =  $(10,\ 3125,\ (\ :33/2, 33/2, 33/2, 33/2, 33/2),\ -1)$
$L(17)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{35}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 10. If $p = 5$, then $F_p$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$ \( ( 1 + p^{16} T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - 3809 p^{3} T + 667764131 p^{5} T^{2} - 148029135825 p^{12} T^{3} + 480463746951481 p^{19} T^{4} - 23561651058556561 p^{28} T^{5} + 480463746951481 p^{52} T^{6} - 148029135825 p^{78} T^{7} + 667764131 p^{104} T^{8} - 3809 p^{135} T^{9} + p^{165} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 4996238 p T + 83201934504583 p^{4} T^{2} - 7840773217846951400 p^{9} T^{3} + \)\(13\!\cdots\!94\)\( p^{15} T^{4} - \)\(54\!\cdots\!32\)\( p^{22} T^{5} + \)\(13\!\cdots\!94\)\( p^{48} T^{6} - 7840773217846951400 p^{75} T^{7} + 83201934504583 p^{103} T^{8} + 4996238 p^{133} T^{9} + p^{165} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 9350365969594 p T + \)\(11\!\cdots\!01\)\( p^{5} T^{2} + \)\(50\!\cdots\!00\)\( p^{3} T^{3} + \)\(22\!\cdots\!02\)\( p^{6} T^{4} + \)\(56\!\cdots\!16\)\( p^{10} T^{5} + \)\(22\!\cdots\!02\)\( p^{39} T^{6} + \)\(50\!\cdots\!00\)\( p^{69} T^{7} + \)\(11\!\cdots\!01\)\( p^{104} T^{8} + 9350365969594 p^{133} T^{9} + p^{165} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 287801801877976640 T + \)\(10\!\cdots\!45\)\( p T^{2} + \)\(19\!\cdots\!80\)\( p^{2} T^{3} + \)\(40\!\cdots\!10\)\( p^{3} T^{4} + \)\(53\!\cdots\!28\)\( p^{4} T^{5} + \)\(40\!\cdots\!10\)\( p^{36} T^{6} + \)\(19\!\cdots\!80\)\( p^{68} T^{7} + \)\(10\!\cdots\!45\)\( p^{100} T^{8} + 287801801877976640 p^{132} T^{9} + p^{165} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 184400088529992882 p T + \)\(15\!\cdots\!81\)\( p T^{2} - \)\(25\!\cdots\!00\)\( p^{2} T^{3} + \)\(85\!\cdots\!94\)\( p^{3} T^{4} - \)\(96\!\cdots\!16\)\( p^{5} T^{5} + \)\(85\!\cdots\!94\)\( p^{36} T^{6} - \)\(25\!\cdots\!00\)\( p^{68} T^{7} + \)\(15\!\cdots\!81\)\( p^{100} T^{8} - 184400088529992882 p^{133} T^{9} + p^{165} T^{10} \)
17$C_2 \wr S_5$ \( 1 + \)\(30\!\cdots\!18\)\( T + \)\(15\!\cdots\!37\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!14\)\( p T^{4} + \)\(61\!\cdots\!56\)\( p^{2} T^{5} + \)\(62\!\cdots\!14\)\( p^{34} T^{6} + \)\(31\!\cdots\!00\)\( p^{66} T^{7} + \)\(15\!\cdots\!37\)\( p^{99} T^{8} + \)\(30\!\cdots\!18\)\( p^{132} T^{9} + p^{165} T^{10} \)
19$C_2 \wr S_5$ \( 1 - \)\(91\!\cdots\!00\)\( T + \)\(53\!\cdots\!95\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!90\)\( p T^{4} - \)\(11\!\cdots\!00\)\( p^{3} T^{5} + \)\(74\!\cdots\!90\)\( p^{34} T^{6} - \)\(37\!\cdots\!00\)\( p^{66} T^{7} + \)\(53\!\cdots\!95\)\( p^{99} T^{8} - \)\(91\!\cdots\!00\)\( p^{132} T^{9} + p^{165} T^{10} \)
23$C_2 \wr S_5$ \( 1 - \)\(55\!\cdots\!46\)\( T + \)\(73\!\cdots\!21\)\( p T^{2} - \)\(10\!\cdots\!00\)\( p^{2} T^{3} + \)\(88\!\cdots\!58\)\( p^{4} T^{4} - \)\(30\!\cdots\!68\)\( p^{4} T^{5} + \)\(88\!\cdots\!58\)\( p^{37} T^{6} - \)\(10\!\cdots\!00\)\( p^{68} T^{7} + \)\(73\!\cdots\!21\)\( p^{100} T^{8} - \)\(55\!\cdots\!46\)\( p^{132} T^{9} + p^{165} T^{10} \)
29$C_2 \wr S_5$ \( 1 + \)\(31\!\cdots\!50\)\( p T + \)\(99\!\cdots\!45\)\( p^{2} T^{2} + \)\(24\!\cdots\!00\)\( p^{3} T^{3} + \)\(40\!\cdots\!10\)\( p^{4} T^{4} + \)\(75\!\cdots\!00\)\( p^{5} T^{5} + \)\(40\!\cdots\!10\)\( p^{37} T^{6} + \)\(24\!\cdots\!00\)\( p^{69} T^{7} + \)\(99\!\cdots\!45\)\( p^{101} T^{8} + \)\(31\!\cdots\!50\)\( p^{133} T^{9} + p^{165} T^{10} \)
31$C_2 \wr S_5$ \( 1 + \)\(13\!\cdots\!40\)\( T + \)\(13\!\cdots\!45\)\( p T^{2} - \)\(17\!\cdots\!20\)\( p^{2} T^{3} + \)\(23\!\cdots\!10\)\( p^{3} T^{4} - \)\(13\!\cdots\!12\)\( p^{4} T^{5} + \)\(23\!\cdots\!10\)\( p^{36} T^{6} - \)\(17\!\cdots\!20\)\( p^{68} T^{7} + \)\(13\!\cdots\!45\)\( p^{100} T^{8} + \)\(13\!\cdots\!40\)\( p^{132} T^{9} + p^{165} T^{10} \)
37$C_2 \wr S_5$ \( 1 + \)\(14\!\cdots\!38\)\( T + \)\(33\!\cdots\!97\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!18\)\( T^{4} + \)\(27\!\cdots\!84\)\( T^{5} + \)\(40\!\cdots\!18\)\( p^{33} T^{6} + \)\(32\!\cdots\!00\)\( p^{66} T^{7} + \)\(33\!\cdots\!97\)\( p^{99} T^{8} + \)\(14\!\cdots\!38\)\( p^{132} T^{9} + p^{165} T^{10} \)
41$C_2 \wr S_5$ \( 1 + \)\(13\!\cdots\!90\)\( T + \)\(13\!\cdots\!45\)\( T^{2} + \)\(98\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!10\)\( T^{4} + \)\(25\!\cdots\!48\)\( T^{5} + \)\(56\!\cdots\!10\)\( p^{33} T^{6} + \)\(98\!\cdots\!80\)\( p^{66} T^{7} + \)\(13\!\cdots\!45\)\( p^{99} T^{8} + \)\(13\!\cdots\!90\)\( p^{132} T^{9} + p^{165} T^{10} \)
43$C_2 \wr S_5$ \( 1 + \)\(23\!\cdots\!94\)\( T + \)\(38\!\cdots\!43\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!98\)\( T^{4} + \)\(55\!\cdots\!12\)\( T^{5} + \)\(56\!\cdots\!98\)\( p^{33} T^{6} + \)\(49\!\cdots\!00\)\( p^{66} T^{7} + \)\(38\!\cdots\!43\)\( p^{99} T^{8} + \)\(23\!\cdots\!94\)\( p^{132} T^{9} + p^{165} T^{10} \)
47$C_2 \wr S_5$ \( 1 + \)\(77\!\cdots\!98\)\( T + \)\(80\!\cdots\!27\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!58\)\( T^{4} + \)\(94\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!58\)\( p^{33} T^{6} + \)\(42\!\cdots\!00\)\( p^{66} T^{7} + \)\(80\!\cdots\!27\)\( p^{99} T^{8} + \)\(77\!\cdots\!98\)\( p^{132} T^{9} + p^{165} T^{10} \)
53$C_2 \wr S_5$ \( 1 - \)\(10\!\cdots\!86\)\( T + \)\(65\!\cdots\!73\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(33\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!58\)\( p^{33} T^{6} - \)\(29\!\cdots\!00\)\( p^{66} T^{7} + \)\(65\!\cdots\!73\)\( p^{99} T^{8} - \)\(10\!\cdots\!86\)\( p^{132} T^{9} + p^{165} T^{10} \)
59$C_2 \wr S_5$ \( 1 - \)\(25\!\cdots\!00\)\( T + \)\(12\!\cdots\!95\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!10\)\( T^{4} - \)\(93\!\cdots\!00\)\( T^{5} + \)\(62\!\cdots\!10\)\( p^{33} T^{6} - \)\(24\!\cdots\!00\)\( p^{66} T^{7} + \)\(12\!\cdots\!95\)\( p^{99} T^{8} - \)\(25\!\cdots\!00\)\( p^{132} T^{9} + p^{165} T^{10} \)
61$C_2 \wr S_5$ \( 1 - \)\(74\!\cdots\!10\)\( T + \)\(40\!\cdots\!45\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!10\)\( T^{4} - \)\(18\!\cdots\!52\)\( T^{5} + \)\(62\!\cdots\!10\)\( p^{33} T^{6} - \)\(17\!\cdots\!20\)\( p^{66} T^{7} + \)\(40\!\cdots\!45\)\( p^{99} T^{8} - \)\(74\!\cdots\!10\)\( p^{132} T^{9} + p^{165} T^{10} \)
67$C_2 \wr S_5$ \( 1 + \)\(37\!\cdots\!18\)\( T + \)\(13\!\cdots\!87\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!38\)\( T^{4} + \)\(81\!\cdots\!84\)\( T^{5} + \)\(58\!\cdots\!38\)\( p^{33} T^{6} + \)\(29\!\cdots\!00\)\( p^{66} T^{7} + \)\(13\!\cdots\!87\)\( p^{99} T^{8} + \)\(37\!\cdots\!18\)\( p^{132} T^{9} + p^{165} T^{10} \)
71$C_2 \wr S_5$ \( 1 + \)\(55\!\cdots\!40\)\( T + \)\(31\!\cdots\!95\)\( T^{2} - \)\(69\!\cdots\!20\)\( T^{3} - \)\(24\!\cdots\!90\)\( T^{4} - \)\(22\!\cdots\!52\)\( T^{5} - \)\(24\!\cdots\!90\)\( p^{33} T^{6} - \)\(69\!\cdots\!20\)\( p^{66} T^{7} + \)\(31\!\cdots\!95\)\( p^{99} T^{8} + \)\(55\!\cdots\!40\)\( p^{132} T^{9} + p^{165} T^{10} \)
73$C_2 \wr S_5$ \( 1 + \)\(13\!\cdots\!54\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(73\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(35\!\cdots\!78\)\( p^{33} T^{6} + \)\(73\!\cdots\!00\)\( p^{66} T^{7} + \)\(13\!\cdots\!33\)\( p^{99} T^{8} + \)\(13\!\cdots\!54\)\( p^{132} T^{9} + p^{165} T^{10} \)
79$C_2 \wr S_5$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(74\!\cdots\!95\)\( T^{2} - \)\(86\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(82\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!10\)\( p^{33} T^{6} - \)\(86\!\cdots\!00\)\( p^{66} T^{7} + \)\(74\!\cdots\!95\)\( p^{99} T^{8} + \)\(10\!\cdots\!00\)\( p^{132} T^{9} + p^{165} T^{10} \)
83$C_2 \wr S_5$ \( 1 - \)\(92\!\cdots\!26\)\( T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(75\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!38\)\( T^{4} - \)\(23\!\cdots\!88\)\( T^{5} + \)\(51\!\cdots\!38\)\( p^{33} T^{6} - \)\(75\!\cdots\!00\)\( p^{66} T^{7} + \)\(11\!\cdots\!63\)\( p^{99} T^{8} - \)\(92\!\cdots\!26\)\( p^{132} T^{9} + p^{165} T^{10} \)
89$C_2 \wr S_5$ \( 1 + \)\(60\!\cdots\!50\)\( T + \)\(56\!\cdots\!45\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!10\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!10\)\( p^{33} T^{6} + \)\(56\!\cdots\!00\)\( p^{66} T^{7} + \)\(56\!\cdots\!45\)\( p^{99} T^{8} + \)\(60\!\cdots\!50\)\( p^{132} T^{9} + p^{165} T^{10} \)
97$C_2 \wr S_5$ \( 1 + \)\(86\!\cdots\!98\)\( T + \)\(15\!\cdots\!77\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} + \)\(62\!\cdots\!84\)\( T^{5} + \)\(10\!\cdots\!58\)\( p^{33} T^{6} + \)\(11\!\cdots\!00\)\( p^{66} T^{7} + \)\(15\!\cdots\!77\)\( p^{99} T^{8} + \)\(86\!\cdots\!98\)\( p^{132} T^{9} + p^{165} T^{10} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.828679055593915378418729351198, −8.705838615392936199684050655286, −8.701726712481703178513907096448, −8.527039804162388425733647181615, −8.400472955148589293271965183543, −7.69301618843731490330940734945, −7.20665897752049466098142899409, −7.18164491004867355002602034511, −6.65713411629284494211826183269, −6.39248164769554864033054971017, −5.72581932017110231922042543767, −5.10432073364808440396615863509, −5.09588363391465376274668579748, −5.06693215451136000491998908408, −4.76194412056382722165578677778, −4.22078326040349388487576387208, −3.82382516576215466991583202151, −3.48760672775751725715664946117, −3.30689421011607447182734749356, −3.29324260045397388126301025346, −2.65789535174190475688200019494, −2.35899316743169770198591914298, −1.59830898057801865036827199718, −1.15229297651253518036906482209, −0.945238004110062940083758615377, 0, 0, 0, 0, 0, 0.945238004110062940083758615377, 1.15229297651253518036906482209, 1.59830898057801865036827199718, 2.35899316743169770198591914298, 2.65789535174190475688200019494, 3.29324260045397388126301025346, 3.30689421011607447182734749356, 3.48760672775751725715664946117, 3.82382516576215466991583202151, 4.22078326040349388487576387208, 4.76194412056382722165578677778, 5.06693215451136000491998908408, 5.09588363391465376274668579748, 5.10432073364808440396615863509, 5.72581932017110231922042543767, 6.39248164769554864033054971017, 6.65713411629284494211826183269, 7.18164491004867355002602034511, 7.20665897752049466098142899409, 7.69301618843731490330940734945, 8.400472955148589293271965183543, 8.527039804162388425733647181615, 8.701726712481703178513907096448, 8.705838615392936199684050655286, 8.828679055593915378418729351198

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