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Properties

Label 6-5e3-1.1-c19e3-0-0
Degree $6$
Conductor $125$
Sign $-1$
Analytic cond. $1497.52$
Root an. cond. $3.38243$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.00e3·2-s − 7.34e4·3-s + 1.97e5·4-s + 5.85e6·5-s + 7.38e7·6-s − 5.49e7·7-s + 1.45e8·8-s + 1.65e9·9-s − 5.89e9·10-s − 8.56e9·11-s − 1.44e10·12-s − 8.56e10·13-s + 5.52e10·14-s − 4.30e11·15-s − 2.55e11·16-s + 8.72e10·17-s − 1.66e12·18-s + 1.28e12·19-s + 1.15e12·20-s + 4.03e12·21-s + 8.61e12·22-s + 4.08e12·23-s − 1.06e13·24-s + 2.28e13·25-s + 8.61e13·26-s + 2.68e12·27-s − 1.08e13·28-s + ⋯
L(s)  = 1  − 1.38·2-s − 2.15·3-s + 0.375·4-s + 1.34·5-s + 2.99·6-s − 0.514·7-s + 0.383·8-s + 1.42·9-s − 1.86·10-s − 1.09·11-s − 0.809·12-s − 2.23·13-s + 0.714·14-s − 2.89·15-s − 0.928·16-s + 0.178·17-s − 1.98·18-s + 0.911·19-s + 0.504·20-s + 1.10·21-s + 1.52·22-s + 0.473·23-s − 0.825·24-s + 6/5·25-s + 3.11·26-s + 0.0676·27-s − 0.193·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+19/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-1$
Analytic conductor: \(1497.52\)
Root analytic conductor: \(3.38243\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 125,\ (\ :19/2, 19/2, 19/2),\ -1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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)$
bad5$C_1$ \( ( 1 - p^{9} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 503 p T + 25469 p^{5} T^{2} + 465055 p^{10} T^{3} + 25469 p^{24} T^{4} + 503 p^{39} T^{5} + p^{57} T^{6} \)
3$S_4\times C_2$ \( 1 + 24484 p T + 15374699 p^{5} T^{2} + 68528004440 p^{7} T^{3} + 15374699 p^{24} T^{4} + 24484 p^{39} T^{5} + p^{57} T^{6} \)
7$S_4\times C_2$ \( 1 + 54910456 T + 3436005091831299 p T^{2} + \)\(25\!\cdots\!00\)\( p^{3} T^{3} + 3436005091831299 p^{20} T^{4} + 54910456 p^{38} T^{5} + p^{57} T^{6} \)
11$S_4\times C_2$ \( 1 + 8566943524 T + \)\(17\!\cdots\!65\)\( T^{2} + \)\(92\!\cdots\!80\)\( p T^{3} + \)\(17\!\cdots\!65\)\( p^{19} T^{4} + 8566943524 p^{38} T^{5} + p^{57} T^{6} \)
13$S_4\times C_2$ \( 1 + 85630509662 T + \)\(64\!\cdots\!07\)\( T^{2} + \)\(19\!\cdots\!20\)\( p T^{3} + \)\(64\!\cdots\!07\)\( p^{19} T^{4} + 85630509662 p^{38} T^{5} + p^{57} T^{6} \)
17$S_4\times C_2$ \( 1 - 87257923094 T + \)\(14\!\cdots\!39\)\( p T^{2} - \)\(36\!\cdots\!60\)\( p^{2} T^{3} + \)\(14\!\cdots\!39\)\( p^{20} T^{4} - 87257923094 p^{38} T^{5} + p^{57} T^{6} \)
19$S_4\times C_2$ \( 1 - 1282010076580 T + \)\(47\!\cdots\!37\)\( T^{2} - \)\(47\!\cdots\!40\)\( T^{3} + \)\(47\!\cdots\!37\)\( p^{19} T^{4} - 1282010076580 p^{38} T^{5} + p^{57} T^{6} \)
23$S_4\times C_2$ \( 1 - 4088829256728 T + \)\(21\!\cdots\!57\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!57\)\( p^{19} T^{4} - 4088829256728 p^{38} T^{5} + p^{57} T^{6} \)
29$S_4\times C_2$ \( 1 + 73280209082030 T + \)\(11\!\cdots\!07\)\( T^{2} + \)\(82\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!07\)\( p^{19} T^{4} + 73280209082030 p^{38} T^{5} + p^{57} T^{6} \)
31$S_4\times C_2$ \( 1 + 284526134418784 T + \)\(67\!\cdots\!65\)\( T^{2} + \)\(97\!\cdots\!80\)\( T^{3} + \)\(67\!\cdots\!65\)\( p^{19} T^{4} + 284526134418784 p^{38} T^{5} + p^{57} T^{6} \)
37$S_4\times C_2$ \( 1 + 11692278250638 p T - \)\(16\!\cdots\!97\)\( T^{2} - \)\(48\!\cdots\!20\)\( T^{3} - \)\(16\!\cdots\!97\)\( p^{19} T^{4} + 11692278250638 p^{39} T^{5} + p^{57} T^{6} \)
41$S_4\times C_2$ \( 1 - 2081433692967886 T + \)\(75\!\cdots\!15\)\( T^{2} - \)\(89\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!15\)\( p^{19} T^{4} - 2081433692967886 p^{38} T^{5} + p^{57} T^{6} \)
43$S_4\times C_2$ \( 1 - 1100001003168508 T + \)\(19\!\cdots\!57\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!57\)\( p^{19} T^{4} - 1100001003168508 p^{38} T^{5} + p^{57} T^{6} \)
47$S_4\times C_2$ \( 1 + 20429801107275856 T + \)\(26\!\cdots\!73\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!73\)\( p^{19} T^{4} + 20429801107275856 p^{38} T^{5} + p^{57} T^{6} \)
53$S_4\times C_2$ \( 1 - 9015898015717898 T + \)\(11\!\cdots\!07\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!07\)\( p^{19} T^{4} - 9015898015717898 p^{38} T^{5} + p^{57} T^{6} \)
59$S_4\times C_2$ \( 1 - 3004038355564940 T + \)\(87\!\cdots\!17\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(87\!\cdots\!17\)\( p^{19} T^{4} - 3004038355564940 p^{38} T^{5} + p^{57} T^{6} \)
61$S_4\times C_2$ \( 1 + 40826301921185774 T + \)\(94\!\cdots\!15\)\( T^{2} + \)\(89\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!15\)\( p^{19} T^{4} + 40826301921185774 p^{38} T^{5} + p^{57} T^{6} \)
67$S_4\times C_2$ \( 1 + 1055285157192202156 T + \)\(51\!\cdots\!13\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(51\!\cdots\!13\)\( p^{19} T^{4} + 1055285157192202156 p^{38} T^{5} + p^{57} T^{6} \)
71$S_4\times C_2$ \( 1 - 295447643020954696 T + \)\(19\!\cdots\!65\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!65\)\( p^{19} T^{4} - 295447643020954696 p^{38} T^{5} + p^{57} T^{6} \)
73$S_4\times C_2$ \( 1 - 481254292945206478 T + \)\(78\!\cdots\!07\)\( T^{2} - \)\(23\!\cdots\!60\)\( T^{3} + \)\(78\!\cdots\!07\)\( p^{19} T^{4} - 481254292945206478 p^{38} T^{5} + p^{57} T^{6} \)
79$S_4\times C_2$ \( 1 - 1996757185645655120 T + \)\(35\!\cdots\!57\)\( T^{2} - \)\(35\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!57\)\( p^{19} T^{4} - 1996757185645655120 p^{38} T^{5} + p^{57} T^{6} \)
83$S_4\times C_2$ \( 1 + 1502750763410367132 T + \)\(27\!\cdots\!57\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!57\)\( p^{19} T^{4} + 1502750763410367132 p^{38} T^{5} + p^{57} T^{6} \)
89$S_4\times C_2$ \( 1 - 1042000807550316510 T + \)\(29\!\cdots\!27\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!27\)\( p^{19} T^{4} - 1042000807550316510 p^{38} T^{5} + p^{57} T^{6} \)
97$S_4\times C_2$ \( 1 - 1619779467732499494 T + \)\(46\!\cdots\!23\)\( T^{2} - \)\(62\!\cdots\!60\)\( T^{3} + \)\(46\!\cdots\!23\)\( p^{19} T^{4} - 1619779467732499494 p^{38} T^{5} + p^{57} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.92792898930972121082970556276, −17.06846030132405874335653820018, −16.92733853764336747891808142553, −16.41745501307115528598446610168, −16.05165878936385602177785957881, −14.77344314025641349310502348618, −14.54841325185355813549005298009, −13.32992663521585378433896349939, −13.10791958330124846812503920470, −12.26361423165621611023465924670, −11.80035455163403749369677636900, −10.83506697928661100763548447710, −10.78632343163000863847494495098, −9.766217182875214417134571829542, −9.557910684594814793278624939829, −9.129250451201655238197164013083, −7.83134970837625860706820684166, −7.24155793277940009029096718617, −6.41375764722231202742940310756, −5.70948491791892528396728684746, −5.11395785348767624181801561293, −4.97693007275000085368321307786, −3.05484137371130244124996458687, −2.18779837217932631972187179474, −1.31457565302715637383500890498, 0, 0, 0, 1.31457565302715637383500890498, 2.18779837217932631972187179474, 3.05484137371130244124996458687, 4.97693007275000085368321307786, 5.11395785348767624181801561293, 5.70948491791892528396728684746, 6.41375764722231202742940310756, 7.24155793277940009029096718617, 7.83134970837625860706820684166, 9.129250451201655238197164013083, 9.557910684594814793278624939829, 9.766217182875214417134571829542, 10.78632343163000863847494495098, 10.83506697928661100763548447710, 11.80035455163403749369677636900, 12.26361423165621611023465924670, 13.10791958330124846812503920470, 13.32992663521585378433896349939, 14.54841325185355813549005298009, 14.77344314025641349310502348618, 16.05165878936385602177785957881, 16.41745501307115528598446610168, 16.92733853764336747891808142553, 17.06846030132405874335653820018, 17.92792898930972121082970556276

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