Properties

Degree 12
Conductor $ 5^{6} $
Sign $1$
Motivic weight 33
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.47e5·2-s + 2.65e7·3-s − 1.53e8·4-s + 9.15e11·5-s + 3.90e12·6-s + 1.91e13·7-s − 4.87e14·8-s − 9.52e15·9-s + 1.34e17·10-s + 1.50e17·11-s − 4.06e15·12-s − 1.40e18·13-s + 2.81e18·14-s + 2.42e19·15-s + 2.07e18·16-s + 1.34e20·17-s − 1.40e21·18-s + 1.71e21·19-s − 1.40e20·20-s + 5.06e20·21-s + 2.22e22·22-s + 6.87e21·23-s − 1.29e22·24-s + 4.88e23·25-s − 2.06e23·26-s − 4.21e23·27-s − 2.93e21·28-s + ⋯
L(s)  = 1  + 1.58·2-s + 0.355·3-s − 0.0178·4-s + 2.68·5-s + 0.565·6-s + 0.217·7-s − 0.612·8-s − 1.71·9-s + 4.26·10-s + 0.989·11-s − 0.00635·12-s − 0.584·13-s + 0.345·14-s + 0.954·15-s + 0.0280·16-s + 0.669·17-s − 2.72·18-s + 1.36·19-s − 0.0479·20-s + 0.0773·21-s + 1.57·22-s + 0.233·23-s − 0.217·24-s + 21/5·25-s − 0.929·26-s − 1.01·27-s − 0.00388·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+33/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(15625\)    =    \(5^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(33\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 15625,\ (\ :[33/2]^{6}),\ 1)$
$L(17)$  $\approx$  $87.51151057$
$L(\frac12)$  $\approx$  $87.51151057$
$L(\frac{35}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 12. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad5 \( ( 1 - p^{16} T )^{6} \)
good2 \( 1 - 73675 p T + 2733186525 p^{3} T^{2} - 21537675951275 p^{7} T^{3} + 5121518228312647 p^{16} T^{4} - 2120910646147229675 p^{24} T^{5} + \)\(15\!\cdots\!25\)\( p^{31} T^{6} - 2120910646147229675 p^{57} T^{7} + 5121518228312647 p^{82} T^{8} - 21537675951275 p^{106} T^{9} + 2733186525 p^{135} T^{10} - 73675 p^{166} T^{11} + p^{198} T^{12} \)
3 \( 1 - 26513900 T + 3409207951532950 p T^{2} - \)\(12\!\cdots\!00\)\( p^{4} T^{3} + \)\(50\!\cdots\!89\)\( p^{9} T^{4} - \)\(60\!\cdots\!00\)\( p^{15} T^{5} + \)\(18\!\cdots\!00\)\( p^{22} T^{6} - \)\(60\!\cdots\!00\)\( p^{48} T^{7} + \)\(50\!\cdots\!89\)\( p^{75} T^{8} - \)\(12\!\cdots\!00\)\( p^{103} T^{9} + 3409207951532950 p^{133} T^{10} - 26513900 p^{165} T^{11} + p^{198} T^{12} \)
7 \( 1 - 19113832847500 T + \)\(38\!\cdots\!50\)\( p T^{2} - \)\(20\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!29\)\( p^{3} T^{4} - \)\(12\!\cdots\!00\)\( p^{6} T^{5} + \)\(12\!\cdots\!00\)\( p^{10} T^{6} - \)\(12\!\cdots\!00\)\( p^{39} T^{7} + \)\(10\!\cdots\!29\)\( p^{69} T^{8} - \)\(20\!\cdots\!00\)\( p^{101} T^{9} + \)\(38\!\cdots\!50\)\( p^{133} T^{10} - 19113832847500 p^{165} T^{11} + p^{198} T^{12} \)
11 \( 1 - 113268892979632 p^{3} T + \)\(62\!\cdots\!26\)\( p^{2} T^{2} - \)\(36\!\cdots\!20\)\( p^{3} T^{3} + \)\(13\!\cdots\!45\)\( p^{5} T^{4} - \)\(59\!\cdots\!92\)\( p^{5} T^{5} + \)\(25\!\cdots\!44\)\( p^{6} T^{6} - \)\(59\!\cdots\!92\)\( p^{38} T^{7} + \)\(13\!\cdots\!45\)\( p^{71} T^{8} - \)\(36\!\cdots\!20\)\( p^{102} T^{9} + \)\(62\!\cdots\!26\)\( p^{134} T^{10} - 113268892979632 p^{168} T^{11} + p^{198} T^{12} \)
13 \( 1 + 1403095636752804700 T + \)\(22\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!00\)\( p T^{3} + \)\(11\!\cdots\!91\)\( p^{3} T^{4} + \)\(57\!\cdots\!00\)\( p^{5} T^{5} + \)\(22\!\cdots\!00\)\( p^{8} T^{6} + \)\(57\!\cdots\!00\)\( p^{38} T^{7} + \)\(11\!\cdots\!91\)\( p^{69} T^{8} + \)\(17\!\cdots\!00\)\( p^{100} T^{9} + \)\(22\!\cdots\!50\)\( p^{132} T^{10} + 1403095636752804700 p^{165} T^{11} + p^{198} T^{12} \)
17 \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(73\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!00\)\( p T^{3} + \)\(83\!\cdots\!63\)\( p^{2} T^{4} + \)\(51\!\cdots\!00\)\( p^{3} T^{5} + \)\(56\!\cdots\!00\)\( p^{4} T^{6} + \)\(51\!\cdots\!00\)\( p^{36} T^{7} + \)\(83\!\cdots\!63\)\( p^{68} T^{8} - \)\(17\!\cdots\!00\)\( p^{100} T^{9} + \)\(73\!\cdots\!50\)\( p^{132} T^{10} - \)\(13\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
19 \( 1 - \)\(17\!\cdots\!00\)\( T + \)\(41\!\cdots\!54\)\( T^{2} - \)\(30\!\cdots\!00\)\( p T^{3} + \)\(33\!\cdots\!15\)\( p^{2} T^{4} - \)\(21\!\cdots\!00\)\( p^{3} T^{5} + \)\(17\!\cdots\!80\)\( p^{4} T^{6} - \)\(21\!\cdots\!00\)\( p^{36} T^{7} + \)\(33\!\cdots\!15\)\( p^{68} T^{8} - \)\(30\!\cdots\!00\)\( p^{100} T^{9} + \)\(41\!\cdots\!54\)\( p^{132} T^{10} - \)\(17\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
23 \( 1 - \)\(29\!\cdots\!00\)\( p T + \)\(22\!\cdots\!50\)\( T^{2} - \)\(89\!\cdots\!00\)\( p T^{3} + \)\(69\!\cdots\!23\)\( p^{2} T^{4} - \)\(23\!\cdots\!00\)\( p^{3} T^{5} + \)\(12\!\cdots\!00\)\( p^{4} T^{6} - \)\(23\!\cdots\!00\)\( p^{36} T^{7} + \)\(69\!\cdots\!23\)\( p^{68} T^{8} - \)\(89\!\cdots\!00\)\( p^{100} T^{9} + \)\(22\!\cdots\!50\)\( p^{132} T^{10} - \)\(29\!\cdots\!00\)\( p^{166} T^{11} + p^{198} T^{12} \)
29 \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(12\!\cdots\!46\)\( p T^{2} - \)\(73\!\cdots\!00\)\( p^{2} T^{3} + \)\(39\!\cdots\!35\)\( p^{3} T^{4} - \)\(25\!\cdots\!00\)\( p^{4} T^{5} + \)\(10\!\cdots\!20\)\( p^{5} T^{6} - \)\(25\!\cdots\!00\)\( p^{37} T^{7} + \)\(39\!\cdots\!35\)\( p^{69} T^{8} - \)\(73\!\cdots\!00\)\( p^{101} T^{9} + \)\(12\!\cdots\!46\)\( p^{133} T^{10} - \)\(13\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
31 \( 1 - \)\(34\!\cdots\!92\)\( p T + \)\(44\!\cdots\!66\)\( p^{3} T^{2} - \)\(30\!\cdots\!20\)\( p^{3} T^{3} + \)\(66\!\cdots\!95\)\( p^{4} T^{4} - \)\(10\!\cdots\!92\)\( p^{5} T^{5} + \)\(15\!\cdots\!84\)\( p^{6} T^{6} - \)\(10\!\cdots\!92\)\( p^{38} T^{7} + \)\(66\!\cdots\!95\)\( p^{70} T^{8} - \)\(30\!\cdots\!20\)\( p^{102} T^{9} + \)\(44\!\cdots\!66\)\( p^{135} T^{10} - \)\(34\!\cdots\!92\)\( p^{166} T^{11} + p^{198} T^{12} \)
37 \( 1 - \)\(27\!\cdots\!00\)\( T + \)\(39\!\cdots\!50\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!27\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(26\!\cdots\!00\)\( p^{33} T^{7} + \)\(31\!\cdots\!27\)\( p^{66} T^{8} - \)\(38\!\cdots\!00\)\( p^{99} T^{9} + \)\(39\!\cdots\!50\)\( p^{132} T^{10} - \)\(27\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
41 \( 1 - \)\(44\!\cdots\!32\)\( T + \)\(40\!\cdots\!86\)\( T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!95\)\( T^{4} - \)\(34\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - \)\(34\!\cdots\!92\)\( p^{33} T^{7} + \)\(95\!\cdots\!95\)\( p^{66} T^{8} - \)\(20\!\cdots\!20\)\( p^{99} T^{9} + \)\(40\!\cdots\!86\)\( p^{132} T^{10} - \)\(44\!\cdots\!32\)\( p^{165} T^{11} + p^{198} T^{12} \)
43 \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(61\!\cdots\!50\)\( T^{2} - \)\(89\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!47\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!00\)\( p^{33} T^{7} + \)\(13\!\cdots\!47\)\( p^{66} T^{8} - \)\(89\!\cdots\!00\)\( p^{99} T^{9} + \)\(61\!\cdots\!50\)\( p^{132} T^{10} - \)\(22\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
47 \( 1 - \)\(38\!\cdots\!00\)\( T + \)\(24\!\cdots\!50\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} - \)\(65\!\cdots\!13\)\( T^{4} + \)\(64\!\cdots\!00\)\( T^{5} - \)\(42\!\cdots\!00\)\( T^{6} + \)\(64\!\cdots\!00\)\( p^{33} T^{7} - \)\(65\!\cdots\!13\)\( p^{66} T^{8} - \)\(21\!\cdots\!00\)\( p^{99} T^{9} + \)\(24\!\cdots\!50\)\( p^{132} T^{10} - \)\(38\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
53 \( 1 - \)\(47\!\cdots\!00\)\( T + \)\(23\!\cdots\!50\)\( T^{2} - \)\(65\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!87\)\( T^{4} - \)\(83\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(83\!\cdots\!00\)\( p^{33} T^{7} + \)\(25\!\cdots\!87\)\( p^{66} T^{8} - \)\(65\!\cdots\!00\)\( p^{99} T^{9} + \)\(23\!\cdots\!50\)\( p^{132} T^{10} - \)\(47\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
59 \( 1 + \)\(17\!\cdots\!00\)\( T + \)\(85\!\cdots\!74\)\( T^{2} + \)\(90\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} + \)\(91\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!80\)\( T^{6} + \)\(91\!\cdots\!00\)\( p^{33} T^{7} + \)\(19\!\cdots\!15\)\( p^{66} T^{8} + \)\(90\!\cdots\!00\)\( p^{99} T^{9} + \)\(85\!\cdots\!74\)\( p^{132} T^{10} + \)\(17\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
61 \( 1 + \)\(28\!\cdots\!08\)\( T + \)\(42\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!08\)\( T^{5} + \)\(85\!\cdots\!84\)\( T^{6} + \)\(16\!\cdots\!08\)\( p^{33} T^{7} + \)\(80\!\cdots\!95\)\( p^{66} T^{8} + \)\(10\!\cdots\!80\)\( p^{99} T^{9} + \)\(42\!\cdots\!46\)\( p^{132} T^{10} + \)\(28\!\cdots\!08\)\( p^{165} T^{11} + p^{198} T^{12} \)
67 \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(90\!\cdots\!50\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!07\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} - \)\(24\!\cdots\!00\)\( p^{33} T^{7} + \)\(22\!\cdots\!07\)\( p^{66} T^{8} - \)\(14\!\cdots\!00\)\( p^{99} T^{9} + \)\(90\!\cdots\!50\)\( p^{132} T^{10} - \)\(33\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
71 \( 1 - \)\(84\!\cdots\!72\)\( T + \)\(11\!\cdots\!06\)\( p T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(99\!\cdots\!92\)\( T^{5} + \)\(39\!\cdots\!44\)\( T^{6} - \)\(99\!\cdots\!92\)\( p^{33} T^{7} + \)\(24\!\cdots\!95\)\( p^{66} T^{8} - \)\(44\!\cdots\!20\)\( p^{99} T^{9} + \)\(11\!\cdots\!06\)\( p^{133} T^{10} - \)\(84\!\cdots\!72\)\( p^{165} T^{11} + p^{198} T^{12} \)
73 \( 1 + \)\(17\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!67\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!00\)\( p^{33} T^{7} + \)\(73\!\cdots\!67\)\( p^{66} T^{8} + \)\(51\!\cdots\!00\)\( p^{99} T^{9} + \)\(12\!\cdots\!50\)\( p^{132} T^{10} + \)\(17\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
79 \( 1 + \)\(22\!\cdots\!00\)\( T + \)\(23\!\cdots\!34\)\( T^{2} + \)\(43\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!15\)\( T^{4} + \)\(34\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(34\!\cdots\!00\)\( p^{33} T^{7} + \)\(23\!\cdots\!15\)\( p^{66} T^{8} + \)\(43\!\cdots\!00\)\( p^{99} T^{9} + \)\(23\!\cdots\!34\)\( p^{132} T^{10} + \)\(22\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
83 \( 1 + \)\(47\!\cdots\!00\)\( T + \)\(88\!\cdots\!50\)\( T^{2} + \)\(37\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!07\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!00\)\( p^{33} T^{7} + \)\(39\!\cdots\!07\)\( p^{66} T^{8} + \)\(37\!\cdots\!00\)\( p^{99} T^{9} + \)\(88\!\cdots\!50\)\( p^{132} T^{10} + \)\(47\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
89 \( 1 + \)\(33\!\cdots\!00\)\( T + \)\(10\!\cdots\!14\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} + \)\(68\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(68\!\cdots\!00\)\( p^{33} T^{7} + \)\(40\!\cdots\!15\)\( p^{66} T^{8} + \)\(20\!\cdots\!00\)\( p^{99} T^{9} + \)\(10\!\cdots\!14\)\( p^{132} T^{10} + \)\(33\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
97 \( 1 - \)\(25\!\cdots\!00\)\( T + \)\(42\!\cdots\!50\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!87\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} - \)\(39\!\cdots\!00\)\( p^{33} T^{7} + \)\(50\!\cdots\!87\)\( p^{66} T^{8} - \)\(52\!\cdots\!00\)\( p^{99} T^{9} + \)\(42\!\cdots\!50\)\( p^{132} T^{10} - \)\(25\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.92367029872052807668853523243, −6.68459592403554418143647784869, −6.36637790558535352432706740019, −5.89530756446408114783769280067, −5.86910312870096593688730688356, −5.71946333803671060531183616557, −5.52324327154557221332631790441, −5.02992339964104360236355512258, −4.75978964464573230976129542578, −4.69396255265465087321603106371, −4.33550553521135272297111188040, −4.10720691939794766588919921973, −3.95397766072631363283846176999, −3.09935720739933656108850924520, −3.02921678763293403068550536557, −2.92267600354425143109178969269, −2.59899542873453747807745214743, −2.44002294241966640068758474033, −1.97942743588026212141340366008, −1.81880437593070529066471579048, −1.33782873674659544208827752433, −0.956507833322988145561840779123, −0.890995480028560626908041798196, −0.66056961295763754977032736231, −0.46818766107997323846354214956, 0.46818766107997323846354214956, 0.66056961295763754977032736231, 0.890995480028560626908041798196, 0.956507833322988145561840779123, 1.33782873674659544208827752433, 1.81880437593070529066471579048, 1.97942743588026212141340366008, 2.44002294241966640068758474033, 2.59899542873453747807745214743, 2.92267600354425143109178969269, 3.02921678763293403068550536557, 3.09935720739933656108850924520, 3.95397766072631363283846176999, 4.10720691939794766588919921973, 4.33550553521135272297111188040, 4.69396255265465087321603106371, 4.75978964464573230976129542578, 5.02992339964104360236355512258, 5.52324327154557221332631790441, 5.71946333803671060531183616557, 5.86910312870096593688730688356, 5.89530756446408114783769280067, 6.36637790558535352432706740019, 6.68459592403554418143647784869, 6.92367029872052807668853523243

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

Plot not available for L-functions of degree greater than 10.