Properties

Label 6-825e3-1.1-c5e3-0-3
Degree $6$
Conductor $561515625$
Sign $-1$
Analytic cond. $2.31655\times 10^{6}$
Root an. cond. $11.5028$
Motivic weight $5$
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  + 7·2-s − 27·3-s − 11·4-s − 189·6-s + 172·7-s − 189·8-s + 486·9-s − 363·11-s + 297·12-s + 654·13-s + 1.20e3·14-s − 427·16-s + 2.36e3·17-s + 3.40e3·18-s − 2.87e3·19-s − 4.64e3·21-s − 2.54e3·22-s − 2.27e3·23-s + 5.10e3·24-s + 4.57e3·26-s − 7.29e3·27-s − 1.89e3·28-s − 7.73e3·29-s + 568·31-s − 1.30e3·32-s + 9.80e3·33-s + 1.65e4·34-s + ⋯
L(s)  = 1  + 1.23·2-s − 1.73·3-s − 0.343·4-s − 2.14·6-s + 1.32·7-s − 1.04·8-s + 2·9-s − 0.904·11-s + 0.595·12-s + 1.07·13-s + 1.64·14-s − 0.416·16-s + 1.98·17-s + 2.47·18-s − 1.82·19-s − 2.29·21-s − 1.11·22-s − 0.895·23-s + 1.80·24-s + 1.32·26-s − 1.92·27-s − 0.456·28-s − 1.70·29-s + 0.106·31-s − 0.225·32-s + 1.56·33-s + 2.45·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(2.31655\times 10^{6}\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 5/2, 5/2, 5/2 ),\ -1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{3} \)
5 \( 1 \)
11$C_1$ \( ( 1 + p^{2} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 7 T + 15 p^{2} T^{2} - 77 p^{2} T^{3} + 15 p^{7} T^{4} - 7 p^{10} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 - 172 T + 53317 T^{2} - 5399144 T^{3} + 53317 p^{5} T^{4} - 172 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 - 654 T + 555091 T^{2} - 167732852 T^{3} + 555091 p^{5} T^{4} - 654 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 - 2366 T + 4754783 T^{2} - 6580937060 T^{3} + 4754783 p^{5} T^{4} - 2366 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 + 2872 T + 9249705 T^{2} + 14234256272 T^{3} + 9249705 p^{5} T^{4} + 2872 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 + 2272 T + 18049957 T^{2} + 25539838016 T^{3} + 18049957 p^{5} T^{4} + 2272 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 + 7738 T + 75886547 T^{2} + 322651167772 T^{3} + 75886547 p^{5} T^{4} + 7738 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 - 568 T + 83955741 T^{2} - 32812503440 T^{3} + 83955741 p^{5} T^{4} - 568 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 - 9126 T + 153307915 T^{2} - 1135693394116 T^{3} + 153307915 p^{5} T^{4} - 9126 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 + 8758 T + 117252903 T^{2} + 906684659284 T^{3} + 117252903 p^{5} T^{4} + 8758 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 14672 T + 370715025 T^{2} - 3794906879008 T^{3} + 370715025 p^{5} T^{4} - 14672 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 19392 T + 652921165 T^{2} - 7386008220288 T^{3} + 652921165 p^{5} T^{4} - 19392 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 - 4598 T + 900328507 T^{2} - 4574622258916 T^{3} + 900328507 p^{5} T^{4} - 4598 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 + 9348 T + 1646289553 T^{2} + 7098388384024 T^{3} + 1646289553 p^{5} T^{4} + 9348 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 + 60078 T + 49584271 p T^{2} + 87982416745556 T^{3} + 49584271 p^{6} T^{4} + 60078 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 - 38468 T + 3866400905 T^{2} - 95393272971992 T^{3} + 3866400905 p^{5} T^{4} - 38468 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 + 74032 T + 6098518645 T^{2} + 250129423986848 T^{3} + 6098518645 p^{5} T^{4} + 74032 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 - 44442 T + 6331091479 T^{2} - 174512379795884 T^{3} + 6331091479 p^{5} T^{4} - 44442 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 + 108116 T + 11158675133 T^{2} + 8537056071080 p T^{3} + 11158675133 p^{5} T^{4} + 108116 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 - 81892 T + 2009956905 T^{2} + 94678226672552 T^{3} + 2009956905 p^{5} T^{4} - 81892 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 167342 T + 25837929495 T^{2} - 2027809825205668 T^{3} + 25837929495 p^{5} T^{4} - 167342 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 + 159702 T + 28145564719 T^{2} + 2389832506953716 T^{3} + 28145564719 p^{5} T^{4} + 159702 p^{10} T^{5} + p^{15} T^{6} \)
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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

−8.923843061495281321639171255843, −8.323033958376440858631661080611, −8.001767243415159885732918871340, −7.914330285723199381888454815096, −7.56837908225641648548646355406, −7.55712335077672776302365328732, −6.76116137485126518151913681885, −6.60856137316361275890159419583, −6.17957841290830233119713451007, −6.02916301198691018755960297134, −5.56116819804269304654440133871, −5.42747433102649607331012164044, −5.40474164098502051732765924007, −4.73376820302772111932277171051, −4.68436205643931953604344196178, −4.45973700874466101902621073428, −3.86024340195301388845383836803, −3.81257051325707501448787147813, −3.66114606219011617983742640275, −2.78456789897390155965773226999, −2.56557075570168019424297531847, −1.91340040158909911062276212887, −1.58627086240070032123947857490, −1.10079465515973933975113574052, −1.09325899895399695595104985822, 0, 0, 0, 1.09325899895399695595104985822, 1.10079465515973933975113574052, 1.58627086240070032123947857490, 1.91340040158909911062276212887, 2.56557075570168019424297531847, 2.78456789897390155965773226999, 3.66114606219011617983742640275, 3.81257051325707501448787147813, 3.86024340195301388845383836803, 4.45973700874466101902621073428, 4.68436205643931953604344196178, 4.73376820302772111932277171051, 5.40474164098502051732765924007, 5.42747433102649607331012164044, 5.56116819804269304654440133871, 6.02916301198691018755960297134, 6.17957841290830233119713451007, 6.60856137316361275890159419583, 6.76116137485126518151913681885, 7.55712335077672776302365328732, 7.56837908225641648548646355406, 7.914330285723199381888454815096, 8.001767243415159885732918871340, 8.323033958376440858631661080611, 8.923843061495281321639171255843

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