Properties

Label 6-45e3-1.1-c17e3-0-2
Degree $6$
Conductor $91125$
Sign $-1$
Analytic cond. $560494.$
Root an. cond. $9.08019$
Motivic weight $17$
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  + 442·2-s + 5.01e4·4-s + 1.17e6·5-s + 4.96e6·7-s + 1.03e7·8-s + 5.17e8·10-s − 1.04e9·11-s − 3.09e9·13-s + 2.19e9·14-s + 1.40e9·16-s + 1.51e10·17-s − 1.11e11·19-s + 5.87e10·20-s − 4.64e11·22-s − 7.84e11·23-s + 9.15e11·25-s − 1.36e12·26-s + 2.48e11·28-s − 1.39e12·29-s − 1.24e13·31-s + 9.24e11·32-s + 6.68e12·34-s + 5.81e12·35-s + 3.20e13·37-s − 4.94e13·38-s + 1.21e13·40-s + 1.02e14·41-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.382·4-s + 1.34·5-s + 0.325·7-s + 0.218·8-s + 1.63·10-s − 1.47·11-s − 1.05·13-s + 0.397·14-s + 0.0817·16-s + 0.525·17-s − 1.51·19-s + 0.513·20-s − 1.80·22-s − 2.08·23-s + 6/5·25-s − 1.28·26-s + 0.124·28-s − 0.517·29-s − 2.62·31-s + 0.148·32-s + 0.641·34-s + 0.436·35-s + 1.50·37-s − 1.84·38-s + 0.292·40-s + 2.00·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91125 ^{s/2} \, \Gamma_{\C}(s+17/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(91125\)    =    \(3^{6} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(560494.\)
Root analytic conductor: \(9.08019\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 91125,\ (\ :17/2, 17/2, 17/2),\ -1)\)

Particular Values

\(L(9)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{19}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - p^{8} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 221 p T + 2269 p^{6} T^{2} - 102281 p^{9} T^{3} + 2269 p^{23} T^{4} - 221 p^{35} T^{5} + p^{51} T^{6} \)
7$S_4\times C_2$ \( 1 - 4962644 T + 1398798051795 p T^{2} + 52159035800223517736 p^{2} T^{3} + 1398798051795 p^{18} T^{4} - 4962644 p^{34} T^{5} + p^{51} T^{6} \)
11$S_4\times C_2$ \( 1 + 1049849720 T + 140907959524085891 p T^{2} + \)\(87\!\cdots\!64\)\( p^{2} T^{3} + 140907959524085891 p^{18} T^{4} + 1049849720 p^{34} T^{5} + p^{51} T^{6} \)
13$S_4\times C_2$ \( 1 + 3091742090 T + 23476900649761178291 T^{2} + \)\(37\!\cdots\!64\)\( p T^{3} + 23476900649761178291 p^{17} T^{4} + 3091742090 p^{34} T^{5} + p^{51} T^{6} \)
17$S_4\times C_2$ \( 1 - 15119940094 T + \)\(79\!\cdots\!43\)\( T^{2} - \)\(44\!\cdots\!68\)\( T^{3} + \)\(79\!\cdots\!43\)\( p^{17} T^{4} - 15119940094 p^{34} T^{5} + p^{51} T^{6} \)
19$S_4\times C_2$ \( 1 + 111896117996 T + \)\(14\!\cdots\!77\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!77\)\( p^{17} T^{4} + 111896117996 p^{34} T^{5} + p^{51} T^{6} \)
23$S_4\times C_2$ \( 1 + 784535528472 T + \)\(53\!\cdots\!85\)\( T^{2} + \)\(21\!\cdots\!12\)\( T^{3} + \)\(53\!\cdots\!85\)\( p^{17} T^{4} + 784535528472 p^{34} T^{5} + p^{51} T^{6} \)
29$S_4\times C_2$ \( 1 + 1395352153678 T + \)\(88\!\cdots\!47\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} + \)\(88\!\cdots\!47\)\( p^{17} T^{4} + 1395352153678 p^{34} T^{5} + p^{51} T^{6} \)
31$S_4\times C_2$ \( 1 + 12458197616536 T + \)\(11\!\cdots\!57\)\( T^{2} + \)\(61\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!57\)\( p^{17} T^{4} + 12458197616536 p^{34} T^{5} + p^{51} T^{6} \)
37$S_4\times C_2$ \( 1 - 32093748310134 T + \)\(67\!\cdots\!95\)\( T^{2} - \)\(11\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!95\)\( p^{17} T^{4} - 32093748310134 p^{34} T^{5} + p^{51} T^{6} \)
41$S_4\times C_2$ \( 1 - 102276917392466 T + \)\(91\!\cdots\!47\)\( T^{2} - \)\(46\!\cdots\!12\)\( T^{3} + \)\(91\!\cdots\!47\)\( p^{17} T^{4} - 102276917392466 p^{34} T^{5} + p^{51} T^{6} \)
43$S_4\times C_2$ \( 1 + 141347515908644 T + \)\(23\!\cdots\!53\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!53\)\( p^{17} T^{4} + 141347515908644 p^{34} T^{5} + p^{51} T^{6} \)
47$S_4\times C_2$ \( 1 - 54182226541768 T + \)\(72\!\cdots\!17\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!17\)\( p^{17} T^{4} - 54182226541768 p^{34} T^{5} + p^{51} T^{6} \)
53$S_4\times C_2$ \( 1 + 345545322785582 T + \)\(52\!\cdots\!15\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(52\!\cdots\!15\)\( p^{17} T^{4} + 345545322785582 p^{34} T^{5} + p^{51} T^{6} \)
59$S_4\times C_2$ \( 1 + 2075534176494536 T + \)\(41\!\cdots\!57\)\( T^{2} + \)\(48\!\cdots\!68\)\( T^{3} + \)\(41\!\cdots\!57\)\( p^{17} T^{4} + 2075534176494536 p^{34} T^{5} + p^{51} T^{6} \)
61$S_4\times C_2$ \( 1 + 1591128188427998 T + \)\(24\!\cdots\!19\)\( T^{2} + \)\(16\!\cdots\!04\)\( T^{3} + \)\(24\!\cdots\!19\)\( p^{17} T^{4} + 1591128188427998 p^{34} T^{5} + p^{51} T^{6} \)
67$S_4\times C_2$ \( 1 + 36085633309132 p T + \)\(26\!\cdots\!93\)\( T^{2} + \)\(56\!\cdots\!68\)\( T^{3} + \)\(26\!\cdots\!93\)\( p^{17} T^{4} + 36085633309132 p^{35} T^{5} + p^{51} T^{6} \)
71$S_4\times C_2$ \( 1 + 5019485593673056 T + \)\(84\!\cdots\!85\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!85\)\( p^{17} T^{4} + 5019485593673056 p^{34} T^{5} + p^{51} T^{6} \)
73$S_4\times C_2$ \( 1 + 10804161755859578 T + \)\(73\!\cdots\!55\)\( T^{2} + \)\(45\!\cdots\!68\)\( T^{3} + \)\(73\!\cdots\!55\)\( p^{17} T^{4} + 10804161755859578 p^{34} T^{5} + p^{51} T^{6} \)
79$S_4\times C_2$ \( 1 + 40642131657181960 T + \)\(88\!\cdots\!77\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(88\!\cdots\!77\)\( p^{17} T^{4} + 40642131657181960 p^{34} T^{5} + p^{51} T^{6} \)
83$S_4\times C_2$ \( 1 - 21823256926416996 T + \)\(10\!\cdots\!49\)\( T^{2} - \)\(14\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!49\)\( p^{17} T^{4} - 21823256926416996 p^{34} T^{5} + p^{51} T^{6} \)
89$S_4\times C_2$ \( 1 - 2700230040372426 T + \)\(34\!\cdots\!47\)\( T^{2} - \)\(45\!\cdots\!08\)\( T^{3} + \)\(34\!\cdots\!47\)\( p^{17} T^{4} - 2700230040372426 p^{34} T^{5} + p^{51} T^{6} \)
97$S_4\times C_2$ \( 1 + 79760576372788314 T + \)\(14\!\cdots\!43\)\( T^{2} + \)\(91\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!43\)\( p^{17} T^{4} + 79760576372788314 p^{34} T^{5} + p^{51} T^{6} \)
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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

−11.26976290627264239422468867938, −10.76479610593183395282742554331, −10.53378436241279286529596135967, −10.32402229981909508670238919969, −9.695986084623531292024287146855, −9.300489624105945855862619210107, −9.232070434372036611233360095158, −8.259212803722348368908240642560, −8.099790279875772905155931126125, −7.46718707117094221168533143583, −7.44614418398470677525399742786, −6.53564985285742875899301739551, −6.28116383351354129468907676233, −5.57716537593172213541321281468, −5.53809666913503281119241491292, −5.33381222980099099059815688829, −4.36019267170279535603251078858, −4.35585629913282994297234904028, −4.19302310159541258492904928685, −3.03101327854685927271895290255, −2.99270835962231112265918300102, −2.26030925502777364569602277058, −2.15213614070625841385683174133, −1.49272126795222372574521703285, −1.38119407959083327441366039144, 0, 0, 0, 1.38119407959083327441366039144, 1.49272126795222372574521703285, 2.15213614070625841385683174133, 2.26030925502777364569602277058, 2.99270835962231112265918300102, 3.03101327854685927271895290255, 4.19302310159541258492904928685, 4.35585629913282994297234904028, 4.36019267170279535603251078858, 5.33381222980099099059815688829, 5.53809666913503281119241491292, 5.57716537593172213541321281468, 6.28116383351354129468907676233, 6.53564985285742875899301739551, 7.44614418398470677525399742786, 7.46718707117094221168533143583, 8.099790279875772905155931126125, 8.259212803722348368908240642560, 9.232070434372036611233360095158, 9.300489624105945855862619210107, 9.695986084623531292024287146855, 10.32402229981909508670238919969, 10.53378436241279286529596135967, 10.76479610593183395282742554331, 11.26976290627264239422468867938

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