Properties

Label 6-825e3-1.1-c5e3-0-1
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 27·3-s − 56·4-s − 54·6-s − 152·7-s + 144·8-s + 486·9-s − 363·11-s − 1.51e3·12-s + 546·13-s + 304·14-s + 1.32e3·16-s + 314·17-s − 972·18-s + 1.80e3·19-s − 4.10e3·21-s + 726·22-s − 4.28e3·23-s + 3.88e3·24-s − 1.09e3·26-s + 7.29e3·27-s + 8.51e3·28-s + 5.58e3·29-s + 6.32e3·31-s − 4.57e3·32-s − 9.80e3·33-s − 628·34-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.73·3-s − 7/4·4-s − 0.612·6-s − 1.17·7-s + 0.795·8-s + 2·9-s − 0.904·11-s − 3.03·12-s + 0.896·13-s + 0.414·14-s + 1.29·16-s + 0.263·17-s − 0.707·18-s + 1.14·19-s − 2.03·21-s + 0.319·22-s − 1.69·23-s + 1.37·24-s − 0.316·26-s + 1.92·27-s + 2.05·28-s + 1.23·29-s + 1.18·31-s − 0.789·32-s − 1.56·33-s − 0.0931·34-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: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(4.011031608\)
\(L(\frac12)\) \(\approx\) \(4.011031608\)
\(L(\frac{7}{2})\) 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$S_4\times C_2$ \( 1 + p T + 15 p^{2} T^{2} + 11 p^{3} T^{3} + 15 p^{7} T^{4} + p^{11} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 + 152 T + 30997 T^{2} + 2145664 T^{3} + 30997 p^{5} T^{4} + 152 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 - 42 p T + 1037131 T^{2} - 397760108 T^{3} + 1037131 p^{5} T^{4} - 42 p^{11} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 - 314 T + 2009423 T^{2} + 187802020 T^{3} + 2009423 p^{5} T^{4} - 314 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 1808 T + 6853545 T^{2} - 8224093888 T^{3} + 6853545 p^{5} T^{4} - 1808 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 + 4288 T + 23159077 T^{2} + 56055432704 T^{3} + 23159077 p^{5} T^{4} + 4288 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 - 5582 T + 67985987 T^{2} - 229315908308 T^{3} + 67985987 p^{5} T^{4} - 5582 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 - 6328 T + 97582461 T^{2} - 367457081360 T^{3} + 97582461 p^{5} T^{4} - 6328 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 16866 T + 247681195 T^{2} + 2094410330156 T^{3} + 247681195 p^{5} T^{4} + 16866 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 23282 T + 380775063 T^{2} - 4449546842396 T^{3} + 380775063 p^{5} T^{4} - 23282 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 + 20572 T + 444556785 T^{2} + 4808229885848 T^{3} + 444556785 p^{5} T^{4} + 20572 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 + 3432 T + 667230445 T^{2} + 1592240231088 T^{3} + 667230445 p^{5} T^{4} + 3432 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 + 16138 T + 921021307 T^{2} + 14483011333916 T^{3} + 921021307 p^{5} T^{4} + 16138 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 - 21972 T + 1997557393 T^{2} - 28848730171256 T^{3} + 1997557393 p^{5} T^{4} - 21972 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 8322 T + 2119997971 T^{2} - 9648987222604 T^{3} + 2119997971 p^{5} T^{4} - 8322 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 - 84332 T + 4880479145 T^{2} - 186562781010248 T^{3} + 4880479145 p^{5} T^{4} - 84332 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 50528 T + 2226337045 T^{2} - 64524591232192 T^{3} + 2226337045 p^{5} T^{4} - 50528 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 - 53838 T + 2994009919 T^{2} - 59248193552036 T^{3} + 2994009919 p^{5} T^{4} - 53838 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 - 6364 T + 8597191613 T^{2} - 39718947883240 T^{3} + 8597191613 p^{5} T^{4} - 6364 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 + 96272 T + 11999711025 T^{2} + 755408965003568 T^{3} + 11999711025 p^{5} T^{4} + 96272 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 + 38938 T + 11007146295 T^{2} + 531082948739852 T^{3} + 11007146295 p^{5} T^{4} + 38938 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 - 103242 T + 15506277199 T^{2} - 1521607808779276 T^{3} + 15506277199 p^{5} T^{4} - 103242 p^{10} T^{5} + p^{15} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.482215120775309968013891966663, −8.102718313577529306531088851740, −8.057116307075131438285682697028, −7.76528065066203192817561623947, −7.25510354805086355741050604983, −7.04229408658193004512435942869, −6.59681478688753380727396214219, −6.43393259516918565755940309837, −5.97113568836800681539075944419, −5.70828310327807472909894539110, −5.03701218988820549792891851670, −4.95152447542382480304315965095, −4.89775317683149519071407058270, −3.99961804413794600935408905284, −3.94079489210083763665820737705, −3.90476392203606472018964619067, −3.21085468562664707665454223682, −3.16457992086396753472234850663, −2.75814671586818336309329274646, −2.36846396987066372542909394937, −1.75103608849350461757826734523, −1.62034891384160577330339635658, −0.843343357606116739670221804741, −0.62003610740880892615285872975, −0.36346721324190111573002025139, 0.36346721324190111573002025139, 0.62003610740880892615285872975, 0.843343357606116739670221804741, 1.62034891384160577330339635658, 1.75103608849350461757826734523, 2.36846396987066372542909394937, 2.75814671586818336309329274646, 3.16457992086396753472234850663, 3.21085468562664707665454223682, 3.90476392203606472018964619067, 3.94079489210083763665820737705, 3.99961804413794600935408905284, 4.89775317683149519071407058270, 4.95152447542382480304315965095, 5.03701218988820549792891851670, 5.70828310327807472909894539110, 5.97113568836800681539075944419, 6.43393259516918565755940309837, 6.59681478688753380727396214219, 7.04229408658193004512435942869, 7.25510354805086355741050604983, 7.76528065066203192817561623947, 8.057116307075131438285682697028, 8.102718313577529306531088851740, 8.482215120775309968013891966663

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