Properties

Label 6-1911e3-1.1-c3e3-0-1
Degree $6$
Conductor $6978821031$
Sign $-1$
Analytic cond. $1.43344\times 10^{6}$
Root an. cond. $10.6185$
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 − 9·3-s − 5·4-s − 4·5-s − 18·6-s − 20·8-s + 54·9-s − 8·10-s − 16·11-s + 45·12-s − 39·13-s + 36·15-s − 51·16-s + 146·17-s + 108·18-s − 94·19-s + 20·20-s − 32·22-s − 48·23-s + 180·24-s − 107·25-s − 78·26-s − 270·27-s − 2·29-s + 72·30-s − 302·31-s + 22·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 5/8·4-s − 0.357·5-s − 1.22·6-s − 0.883·8-s + 2·9-s − 0.252·10-s − 0.438·11-s + 1.08·12-s − 0.832·13-s + 0.619·15-s − 0.796·16-s + 2.08·17-s + 1.41·18-s − 1.13·19-s + 0.223·20-s − 0.310·22-s − 0.435·23-s + 1.53·24-s − 0.855·25-s − 0.588·26-s − 1.92·27-s − 0.0128·29-s + 0.438·30-s − 1.74·31-s + 0.121·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 7^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(1.43344\times 10^{6}\)
Root analytic conductor: \(10.6185\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 7^{6} \cdot 13^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

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$C_1$ \( ( 1 + p T )^{3} \)
7 \( 1 \)
13$C_1$ \( ( 1 + p T )^{3} \)
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$S_4\times C_2$ \( 1 + 4 T + 123 T^{2} + 1864 T^{3} + 123 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \)
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

−8.081710562578517393354975868865, −7.87379015322877909871182323826, −7.82125772884508742186192723281, −7.20349507018252834516221930495, −7.02845403045868285285897524642, −6.93557434518788941437290094591, −6.44823656165554871217983242200, −6.36851349271331545255698142678, −5.86451817394484807394931397942, −5.64868701587967349255626743914, −5.38872743010838534126955899405, −5.28386788146985013319237749891, −5.16925285792939813064254167304, −4.50232464150569206248988607496, −4.41166913500724939127220877669, −4.27423683417682693991271648901, −3.83154685069192198182006100195, −3.57727119365392231159850898289, −3.23737000721425224917404462361, −2.83784662651000038903357195184, −2.40441359621506760787254472720, −1.86065915165469368728982489823, −1.75036970174001650709427511561, −1.08363286837777983982960377311, −0.78275453594945253044368627680, 0, 0, 0, 0.78275453594945253044368627680, 1.08363286837777983982960377311, 1.75036970174001650709427511561, 1.86065915165469368728982489823, 2.40441359621506760787254472720, 2.83784662651000038903357195184, 3.23737000721425224917404462361, 3.57727119365392231159850898289, 3.83154685069192198182006100195, 4.27423683417682693991271648901, 4.41166913500724939127220877669, 4.50232464150569206248988607496, 5.16925285792939813064254167304, 5.28386788146985013319237749891, 5.38872743010838534126955899405, 5.64868701587967349255626743914, 5.86451817394484807394931397942, 6.36851349271331545255698142678, 6.44823656165554871217983242200, 6.93557434518788941437290094591, 7.02845403045868285285897524642, 7.20349507018252834516221930495, 7.82125772884508742186192723281, 7.87379015322877909871182323826, 8.081710562578517393354975868865

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