Properties

Label 6-165e3-1.1-c5e3-0-4
Degree $6$
Conductor $4492125$
Sign $-1$
Analytic cond. $18532.4$
Root an. cond. $5.14425$
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 + 75·5-s − 189·6-s − 172·7-s + 189·8-s + 486·9-s − 525·10-s − 363·11-s − 297·12-s − 654·13-s + 1.20e3·14-s + 2.02e3·15-s − 427·16-s − 2.36e3·17-s − 3.40e3·18-s − 2.87e3·19-s − 825·20-s − 4.64e3·21-s + 2.54e3·22-s + 2.27e3·23-s + 5.10e3·24-s + 3.75e3·25-s + 4.57e3·26-s + 7.29e3·27-s + 1.89e3·28-s + ⋯
L(s)  = 1  − 1.23·2-s + 1.73·3-s − 0.343·4-s + 1.34·5-s − 2.14·6-s − 1.32·7-s + 1.04·8-s + 2·9-s − 1.66·10-s − 0.904·11-s − 0.595·12-s − 1.07·13-s + 1.64·14-s + 2.32·15-s − 0.416·16-s − 1.98·17-s − 2.47·18-s − 1.82·19-s − 0.461·20-s − 2.29·21-s + 1.11·22-s + 0.895·23-s + 1.80·24-s + 6/5·25-s + 1.32·26-s + 1.92·27-s + 0.456·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(4492125\)    =    \(3^{3} \cdot 5^{3} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(18532.4\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 4492125,\ (\ :5/2, 5/2, 5/2),\ -1)\)

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$C_1$ \( ( 1 - p^{2} T )^{3} \)
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

−10.95234006513408669213722949422, −10.22529094599639638795276945044, −10.18133385577831479801716202034, −10.11555758995383618390600633519, −9.466133730767755497954006892179, −9.369676587058756600020413038646, −9.068923594746631202902494570613, −8.779807297412496253561762992226, −8.448841756819302632539310932998, −8.446637262097494723882953486691, −7.58765644839194753908363378024, −7.40374916090889450428123575068, −6.94196864888982131119391313733, −6.60430373285099731449965007460, −6.14326541012814161427172051768, −5.87617634421056300436678483836, −4.81813386704793197314484513086, −4.66526059050335581442195082675, −4.63953215893348607838763329984, −3.46188667418285481120055372018, −3.36354315767284527404667009527, −2.69442915745644190984397759167, −2.35840565968180940761034729307, −1.68731680056053354668745405550, −1.65207071322074254971014894369, 0, 0, 0, 1.65207071322074254971014894369, 1.68731680056053354668745405550, 2.35840565968180940761034729307, 2.69442915745644190984397759167, 3.36354315767284527404667009527, 3.46188667418285481120055372018, 4.63953215893348607838763329984, 4.66526059050335581442195082675, 4.81813386704793197314484513086, 5.87617634421056300436678483836, 6.14326541012814161427172051768, 6.60430373285099731449965007460, 6.94196864888982131119391313733, 7.40374916090889450428123575068, 7.58765644839194753908363378024, 8.446637262097494723882953486691, 8.448841756819302632539310932998, 8.779807297412496253561762992226, 9.068923594746631202902494570613, 9.369676587058756600020413038646, 9.466133730767755497954006892179, 10.11555758995383618390600633519, 10.18133385577831479801716202034, 10.22529094599639638795276945044, 10.95234006513408669213722949422

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