Properties

Label 6-39e6-1.1-c3e3-0-0
Degree $6$
Conductor $3518743761$
Sign $-1$
Analytic cond. $722746.$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5·4-s + 4·5-s − 30·7-s − 20·8-s + 8·10-s − 16·11-s − 60·14-s − 51·16-s + 146·17-s − 94·19-s − 20·20-s − 32·22-s + 48·23-s − 107·25-s + 150·28-s + 2·29-s − 302·31-s + 22·32-s + 292·34-s − 120·35-s − 374·37-s − 188·38-s − 80·40-s + 480·41-s − 260·43-s + 80·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 5/8·4-s + 0.357·5-s − 1.61·7-s − 0.883·8-s + 0.252·10-s − 0.438·11-s − 1.14·14-s − 0.796·16-s + 2.08·17-s − 1.13·19-s − 0.223·20-s − 0.310·22-s + 0.435·23-s − 0.855·25-s + 1.01·28-s + 0.0128·29-s − 1.74·31-s + 0.121·32-s + 1.47·34-s − 0.579·35-s − 1.66·37-s − 0.802·38-s − 0.316·40-s + 1.82·41-s − 0.922·43-s + 0.274·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(722746.\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{6} \cdot 13^{6} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$S_4\times C_2$ \( 1 - p T + 9 T^{2} - p^{3} T^{3} + 9 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \)
5$S_4\times C_2$ \( 1 - 4 T + 123 T^{2} - 1864 T^{3} + 123 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 30 T + 741 T^{2} + 18596 T^{3} + 741 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 16 T + 1737 T^{2} + 72928 T^{3} + 1737 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 146 T + 20799 T^{2} - 1505852 T^{3} + 20799 p^{3} T^{4} - 146 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 94 T + 6145 T^{2} + 509876 T^{3} + 6145 p^{3} T^{4} + 94 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 48 T + 15573 T^{2} - 1702560 T^{3} + 15573 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 63051 T^{2} + 101620 T^{3} + 63051 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 302 T + 71837 T^{2} + 10796516 T^{3} + 71837 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 374 T + 114995 T^{2} + 30130340 T^{3} + 114995 p^{3} T^{4} + 374 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 480 T + 208479 T^{2} - 53244336 T^{3} + 208479 p^{3} T^{4} - 480 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 260 T + 200425 T^{2} + 37680472 T^{3} + 200425 p^{3} T^{4} + 260 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 24 T + 142989 T^{2} + 23086032 T^{3} + 142989 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 678 T + 404403 T^{2} - 200405604 T^{3} + 404403 p^{3} T^{4} - 678 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 1788 T + 1572249 T^{2} + 871859112 T^{3} + 1572249 p^{3} T^{4} + 1788 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 230 T + 636491 T^{2} - 98131748 T^{3} + 636491 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 74 T + 493073 T^{2} + 40252028 T^{3} + 493073 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 948 T + 1061157 T^{2} + 608134872 T^{3} + 1061157 p^{3} T^{4} + 948 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 222 T + 223815 T^{2} - 195504100 T^{3} + 223815 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 24 T + 1400781 T^{2} + 31423696 T^{3} + 1400781 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 796 T + 1904433 T^{2} + 924248872 T^{3} + 1904433 p^{3} T^{4} + 796 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 1436 T + 2536191 T^{2} - 2054800856 T^{3} + 2536191 p^{3} T^{4} - 1436 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 3242 T + 6203519 T^{2} + 7136252780 T^{3} + 6203519 p^{3} T^{4} + 3242 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.629115461214584515797831362494, −7.987193047777157911475014431801, −7.958816907261189542599942445383, −7.70017653773330940899221171091, −7.28602911265057059055142842483, −6.92639283346978393435394052734, −6.88573615103041936550465959978, −6.46240351099753395076800363429, −6.12633880183201989676191793691, −5.78758474154498648997211127016, −5.78042120064467830866333698018, −5.45251695268988356899321811826, −5.11821223993511495805119607344, −4.84698830675110083589147961760, −4.38352851836471252705677979289, −4.18927408566201933598434553611, −3.82613937378273430647774141373, −3.46933826248398265101768088614, −3.36525973481576649496394040510, −2.89990960219841569220249167622, −2.68097238726124859097285883993, −2.28156326127550384971913573599, −1.64012638054825888644664733445, −1.46550902759585952670886832760, −0.873477208503307636792463945465, 0, 0, 0, 0.873477208503307636792463945465, 1.46550902759585952670886832760, 1.64012638054825888644664733445, 2.28156326127550384971913573599, 2.68097238726124859097285883993, 2.89990960219841569220249167622, 3.36525973481576649496394040510, 3.46933826248398265101768088614, 3.82613937378273430647774141373, 4.18927408566201933598434553611, 4.38352851836471252705677979289, 4.84698830675110083589147961760, 5.11821223993511495805119607344, 5.45251695268988356899321811826, 5.78042120064467830866333698018, 5.78758474154498648997211127016, 6.12633880183201989676191793691, 6.46240351099753395076800363429, 6.88573615103041936550465959978, 6.92639283346978393435394052734, 7.28602911265057059055142842483, 7.70017653773330940899221171091, 7.958816907261189542599942445383, 7.987193047777157911475014431801, 8.629115461214584515797831362494

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