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Properties

Label 6-5e3-1.1-c13e3-0-0
Degree $6$
Conductor $125$
Sign $1$
Analytic cond. $154.123$
Root an. cond. $2.31550$
Motivic weight $13$
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  + 142·2-s + 416·3-s + 6.73e3·4-s + 4.68e4·5-s + 5.90e4·6-s + 4.48e5·7-s + 4.73e5·8-s − 1.66e6·9-s + 6.65e6·10-s − 6.60e6·11-s + 2.80e6·12-s − 3.35e7·13-s + 6.36e7·14-s + 1.95e7·15-s + 6.92e7·16-s + 8.31e7·17-s − 2.35e8·18-s + 9.74e7·19-s + 3.15e8·20-s + 1.86e8·21-s − 9.37e8·22-s + 3.16e8·23-s + 1.97e8·24-s + 1.46e9·25-s − 4.75e9·26-s + 1.23e9·27-s + 3.01e9·28-s + ⋯
L(s)  = 1  + 1.56·2-s + 0.329·3-s + 0.821·4-s + 1.34·5-s + 0.516·6-s + 1.44·7-s + 0.639·8-s − 1.04·9-s + 2.10·10-s − 1.12·11-s + 0.270·12-s − 1.92·13-s + 2.25·14-s + 0.442·15-s + 1.03·16-s + 0.835·17-s − 1.63·18-s + 0.475·19-s + 1.10·20-s + 0.474·21-s − 1.76·22-s + 0.445·23-s + 0.210·24-s + 6/5·25-s − 3.02·26-s + 0.614·27-s + 1.18·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+13/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $1$
Analytic conductor: \(154.123\)
Root analytic conductor: \(2.31550\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 125,\ (\ :13/2, 13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(8.597592666\)
\(L(\frac12)\) \(\approx\) \(8.597592666\)
\(L(\frac{15}{2})\) 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$C_1$ \( ( 1 - p^{6} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 71 p T + 1679 p^{3} T^{2} - 11135 p^{7} T^{3} + 1679 p^{16} T^{4} - 71 p^{27} T^{5} + p^{39} T^{6} \)
3$S_4\times C_2$ \( 1 - 416 T + 611651 p T^{2} - 33225280 p^{4} T^{3} + 611651 p^{14} T^{4} - 416 p^{26} T^{5} + p^{39} T^{6} \)
7$S_4\times C_2$ \( 1 - 448292 T + 265294837557 T^{2} - 9371505504195400 p T^{3} + 265294837557 p^{13} T^{4} - 448292 p^{26} T^{5} + p^{39} T^{6} \)
11$S_4\times C_2$ \( 1 + 600364 p T + 967991042665 p^{2} T^{2} + 349257054550580680 p^{3} T^{3} + 967991042665 p^{15} T^{4} + 600364 p^{27} T^{5} + p^{39} T^{6} \)
13$S_4\times C_2$ \( 1 + 33501974 T + 908837738845763 T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + 908837738845763 p^{13} T^{4} + 33501974 p^{26} T^{5} + p^{39} T^{6} \)
17$S_4\times C_2$ \( 1 - 83129542 T + 15503955856788607 T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + 15503955856788607 p^{13} T^{4} - 83129542 p^{26} T^{5} + p^{39} T^{6} \)
19$S_4\times C_2$ \( 1 - 97491100 T + 62056832238727577 T^{2} - \)\(92\!\cdots\!00\)\( T^{3} + 62056832238727577 p^{13} T^{4} - 97491100 p^{26} T^{5} + p^{39} T^{6} \)
23$S_4\times C_2$ \( 1 - 316255836 T + 30454663727526051 p T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + 30454663727526051 p^{14} T^{4} - 316255836 p^{26} T^{5} + p^{39} T^{6} \)
29$S_4\times C_2$ \( 1 - 2236171850 T + 22868957480011100467 T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + 22868957480011100467 p^{13} T^{4} - 2236171850 p^{26} T^{5} + p^{39} T^{6} \)
31$S_4\times C_2$ \( 1 - 7482994376 T + 71432250099037817565 T^{2} - \)\(35\!\cdots\!20\)\( T^{3} + 71432250099037817565 p^{13} T^{4} - 7482994376 p^{26} T^{5} + p^{39} T^{6} \)
37$S_4\times C_2$ \( 1 - 31447174242 T + \)\(72\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(72\!\cdots\!07\)\( p^{13} T^{4} - 31447174242 p^{26} T^{5} + p^{39} T^{6} \)
41$S_4\times C_2$ \( 1 + 262265474 p T + \)\(89\!\cdots\!15\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(89\!\cdots\!15\)\( p^{13} T^{4} + 262265474 p^{27} T^{5} + p^{39} T^{6} \)
43$S_4\times C_2$ \( 1 - 16930554856 T + \)\(50\!\cdots\!93\)\( T^{2} - \)\(57\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!93\)\( p^{13} T^{4} - 16930554856 p^{26} T^{5} + p^{39} T^{6} \)
47$S_4\times C_2$ \( 1 - 31934201692 T + \)\(98\!\cdots\!57\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(98\!\cdots\!57\)\( p^{13} T^{4} - 31934201692 p^{26} T^{5} + p^{39} T^{6} \)
53$S_4\times C_2$ \( 1 + 221149123934 T + \)\(62\!\cdots\!03\)\( T^{2} + \)\(82\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!03\)\( p^{13} T^{4} + 221149123934 p^{26} T^{5} + p^{39} T^{6} \)
59$S_4\times C_2$ \( 1 + 55436423900 T + \)\(70\!\cdots\!37\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!37\)\( p^{13} T^{4} + 55436423900 p^{26} T^{5} + p^{39} T^{6} \)
61$S_4\times C_2$ \( 1 - 496161392746 T + \)\(31\!\cdots\!15\)\( T^{2} - \)\(76\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!15\)\( p^{13} T^{4} - 496161392746 p^{26} T^{5} + p^{39} T^{6} \)
67$S_4\times C_2$ \( 1 - 459297824792 T + \)\(16\!\cdots\!57\)\( T^{2} - \)\(49\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!57\)\( p^{13} T^{4} - 459297824792 p^{26} T^{5} + p^{39} T^{6} \)
71$S_4\times C_2$ \( 1 - 521997878336 T + \)\(21\!\cdots\!65\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(21\!\cdots\!65\)\( p^{13} T^{4} - 521997878336 p^{26} T^{5} + p^{39} T^{6} \)
73$S_4\times C_2$ \( 1 - 34315418782 p T + \)\(58\!\cdots\!23\)\( T^{2} - \)\(82\!\cdots\!40\)\( T^{3} + \)\(58\!\cdots\!23\)\( p^{13} T^{4} - 34315418782 p^{27} T^{5} + p^{39} T^{6} \)
79$S_4\times C_2$ \( 1 - 2990636883200 T + \)\(14\!\cdots\!17\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!17\)\( p^{13} T^{4} - 2990636883200 p^{26} T^{5} + p^{39} T^{6} \)
83$S_4\times C_2$ \( 1 - 5137135467696 T + \)\(32\!\cdots\!33\)\( T^{2} - \)\(89\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!33\)\( p^{13} T^{4} - 5137135467696 p^{26} T^{5} + p^{39} T^{6} \)
89$S_4\times C_2$ \( 1 + 19423025958450 T + \)\(18\!\cdots\!07\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!07\)\( p^{13} T^{4} + 19423025958450 p^{26} T^{5} + p^{39} T^{6} \)
97$S_4\times C_2$ \( 1 + 11088325396458 T + \)\(15\!\cdots\!07\)\( T^{2} + \)\(92\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!07\)\( p^{13} T^{4} + 11088325396458 p^{26} T^{5} + p^{39} T^{6} \)
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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

−18.51971817612987176622737764866, −17.74095729215255639128039610725, −17.63728700278743123019918404002, −16.89973232271445490786378578041, −16.53073923077725540493031590639, −15.37657131552316998790642108833, −14.80976008223161529741057830248, −14.28036333110262779137993025988, −14.02242863159591387673483780041, −13.79472013632195921602640746316, −12.80579314586801838878171470279, −12.51886311031713670285943913899, −11.59458576147576854004784967965, −10.91257459402900938674352492046, −10.02488681619981063906048621245, −9.578739638157912096536922881757, −8.053295773867489441672942657162, −8.010002673231643548827526324616, −6.57602790135658685973759854778, −5.34444984083755281161246773937, −5.10302310791257116066803665336, −4.58968133485545444703984261478, −2.79092381061213603980316841100, −2.45800016280688572336685159833, −1.05916725719876901320017147858, 1.05916725719876901320017147858, 2.45800016280688572336685159833, 2.79092381061213603980316841100, 4.58968133485545444703984261478, 5.10302310791257116066803665336, 5.34444984083755281161246773937, 6.57602790135658685973759854778, 8.010002673231643548827526324616, 8.053295773867489441672942657162, 9.578739638157912096536922881757, 10.02488681619981063906048621245, 10.91257459402900938674352492046, 11.59458576147576854004784967965, 12.51886311031713670285943913899, 12.80579314586801838878171470279, 13.79472013632195921602640746316, 14.02242863159591387673483780041, 14.28036333110262779137993025988, 14.80976008223161529741057830248, 15.37657131552316998790642108833, 16.53073923077725540493031590639, 16.89973232271445490786378578041, 17.63728700278743123019918404002, 17.74095729215255639128039610725, 18.51971817612987176622737764866

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