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Properties

Label 6-175e3-1.1-c9e3-0-0
Degree $6$
Conductor $5359375$
Sign $-1$
Analytic cond. $732194.$
Root an. cond. $9.49374$
Motivic weight $9$
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  − 21·2-s − 84·3-s + 231·4-s + 1.76e3·6-s − 7.20e3·7-s − 2.86e3·8-s − 3.89e4·9-s − 3.44e3·11-s − 1.94e4·12-s + 1.97e4·13-s + 1.51e5·14-s − 4.49e4·16-s − 1.01e6·17-s + 8.18e5·18-s + 2.22e5·19-s + 6.05e5·21-s + 7.23e4·22-s − 1.88e6·23-s + 2.40e5·24-s − 4.15e5·26-s + 3.84e6·27-s − 1.66e6·28-s + 4.08e6·29-s + 2.86e6·31-s − 2.26e6·32-s + 2.89e5·33-s + 2.13e7·34-s + ⋯
L(s)  = 1  − 0.928·2-s − 0.598·3-s + 0.451·4-s + 0.555·6-s − 1.13·7-s − 0.247·8-s − 1.98·9-s − 0.0709·11-s − 0.270·12-s + 0.192·13-s + 1.05·14-s − 0.171·16-s − 2.95·17-s + 1.83·18-s + 0.392·19-s + 0.678·21-s + 0.0658·22-s − 1.40·23-s + 0.148·24-s − 0.178·26-s + 1.39·27-s − 0.511·28-s + 1.07·29-s + 0.558·31-s − 0.381·32-s + 0.0424·33-s + 2.74·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5359375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5359375 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(5359375\)    =    \(5^{6} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(732194.\)
Root analytic conductor: \(9.49374\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 5359375,\ (\ :9/2, 9/2, 9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p^{4} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 21 T + 105 p T^{2} + 303 p^{3} T^{3} + 105 p^{10} T^{4} + 21 p^{18} T^{5} + p^{27} T^{6} \)
3$S_4\times C_2$ \( 1 + 28 p T + 5117 p^{2} T^{2} + 122360 p^{3} T^{3} + 5117 p^{11} T^{4} + 28 p^{19} T^{5} + p^{27} T^{6} \)
11$S_4\times C_2$ \( 1 + 3444 T + 455343105 T^{2} + 125101303155960 T^{3} + 455343105 p^{9} T^{4} + 3444 p^{18} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 - 19782 T + 18882215055 T^{2} - 378008000651932 T^{3} + 18882215055 p^{9} T^{4} - 19782 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 + 1016694 T + 649258140783 T^{2} + 263109059935862868 T^{3} + 649258140783 p^{9} T^{4} + 1016694 p^{18} T^{5} + p^{27} T^{6} \)
19$S_4\times C_2$ \( 1 - 222852 T + 614081373717 T^{2} - 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} - 222852 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 + 81984 p T + 5417652680517 T^{2} + 5817973976209389696 T^{3} + 5417652680517 p^{9} T^{4} + 81984 p^{19} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 - 4081818 T + 38739015783987 T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + 38739015783987 p^{9} T^{4} - 4081818 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 - 2869440 T + 21142500166221 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + 21142500166221 p^{9} T^{4} - 2869440 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 + 1395618 T + 262675972194027 T^{2} + \)\(70\!\cdots\!00\)\( T^{3} + 262675972194027 p^{9} T^{4} + 1395618 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 + 14420658 T + 764979654799959 T^{2} + \)\(74\!\cdots\!64\)\( T^{3} + 764979654799959 p^{9} T^{4} + 14420658 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 - 61631172 T + 2687603003165025 T^{2} - \)\(16\!\cdots\!04\)\( p T^{3} + 2687603003165025 p^{9} T^{4} - 61631172 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 - 10368960 T + 2946826961339709 T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + 2946826961339709 p^{9} T^{4} - 10368960 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 + 67502610 T + 6295046710287531 T^{2} + \)\(20\!\cdots\!32\)\( T^{3} + 6295046710287531 p^{9} T^{4} + 67502610 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 + 42590100 T + 19076976504365997 T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + 19076976504365997 p^{9} T^{4} + 42590100 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 - 191746842 T + 38596678668907359 T^{2} - \)\(44\!\cdots\!36\)\( T^{3} + 38596678668907359 p^{9} T^{4} - 191746842 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 - 255175788 T + 80172518705654361 T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + 80172518705654361 p^{9} T^{4} - 255175788 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 - 296514504 T + 147895725194380437 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + 147895725194380437 p^{9} T^{4} - 296514504 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 + 344213310 T + 83298340302082311 T^{2} + \)\(20\!\cdots\!12\)\( T^{3} + 83298340302082311 p^{9} T^{4} + 344213310 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 + 960412656 T + 566786434394061357 T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + 566786434394061357 p^{9} T^{4} + 960412656 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 - 1100517180 T + 873896700882341301 T^{2} - \)\(43\!\cdots\!28\)\( T^{3} + 873896700882341301 p^{9} T^{4} - 1100517180 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 - 506816478 T + 956194525794688887 T^{2} - \)\(35\!\cdots\!64\)\( T^{3} + 956194525794688887 p^{9} T^{4} - 506816478 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 - 647498250 T + 819696244591424799 T^{2} - \)\(48\!\cdots\!84\)\( T^{3} + 819696244591424799 p^{9} T^{4} - 647498250 p^{18} T^{5} + p^{27} T^{6} \)
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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

−10.37834716956538260633371184062, −9.788530739794358081501145898745, −9.344772685129490715066985952676, −9.283465067646399912760846369696, −8.727435038290865487068702517424, −8.632978553807167871806733239267, −8.452508743579629956172730432645, −7.76428635774332607700997674421, −7.61320621028364699328295100302, −6.91870475751686756886432430136, −6.64343944577379389018788736492, −6.29482439980768693608136196393, −6.18725511061410202033328038541, −5.76992256572035306271007633339, −5.27484935332572328204873672872, −4.93920056826668455712181757304, −4.32909173481365273822846362323, −4.02181651458375474891486032350, −3.51323292606239476890250889953, −2.99431502674022672929511749371, −2.54732559978422719794758253914, −2.27953042054204630079114654229, −2.05349547033056550107787523715, −1.05183241118828760893794326512, −0.76588472468395388654175095457, 0, 0, 0, 0.76588472468395388654175095457, 1.05183241118828760893794326512, 2.05349547033056550107787523715, 2.27953042054204630079114654229, 2.54732559978422719794758253914, 2.99431502674022672929511749371, 3.51323292606239476890250889953, 4.02181651458375474891486032350, 4.32909173481365273822846362323, 4.93920056826668455712181757304, 5.27484935332572328204873672872, 5.76992256572035306271007633339, 6.18725511061410202033328038541, 6.29482439980768693608136196393, 6.64343944577379389018788736492, 6.91870475751686756886432430136, 7.61320621028364699328295100302, 7.76428635774332607700997674421, 8.452508743579629956172730432645, 8.632978553807167871806733239267, 8.727435038290865487068702517424, 9.283465067646399912760846369696, 9.344772685129490715066985952676, 9.788530739794358081501145898745, 10.37834716956538260633371184062

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