Properties

Label 6-165e3-1.1-c5e3-0-1
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s − 27·3-s + 13·4-s − 75·5-s − 189·6-s + 92·7-s − 21·8-s + 486·9-s − 525·10-s + 363·11-s − 351·12-s − 90·13-s + 644·14-s + 2.02e3·15-s − 787·16-s + 1.93e3·17-s + 3.40e3·18-s + 2.08e3·19-s − 975·20-s − 2.48e3·21-s + 2.54e3·22-s + 1.22e3·23-s + 567·24-s + 3.75e3·25-s − 630·26-s − 7.29e3·27-s + 1.19e3·28-s + ⋯
L(s)  = 1  + 1.23·2-s − 1.73·3-s + 0.406·4-s − 1.34·5-s − 2.14·6-s + 0.709·7-s − 0.116·8-s + 2·9-s − 1.66·10-s + 0.904·11-s − 0.703·12-s − 0.147·13-s + 0.878·14-s + 2.32·15-s − 0.768·16-s + 1.62·17-s + 2.47·18-s + 1.32·19-s − 0.545·20-s − 1.22·21-s + 1.11·22-s + 0.480·23-s + 0.200·24-s + 6/5·25-s − 0.182·26-s − 1.92·27-s + 0.288·28-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: \(0\)
Selberg data: \((6,\ 4492125,\ (\ :5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.141019840\)
\(L(\frac12)\) \(\approx\) \(4.141019840\)
\(L(\frac{7}{2})\) 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$S_4\times C_2$ \( 1 - 7 T + 9 p^{2} T^{2} - 35 p^{2} T^{3} + 9 p^{7} T^{4} - 7 p^{10} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 - 92 T + 28489 T^{2} - 1066120 T^{3} + 28489 p^{5} T^{4} - 92 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 + 90 T + 138199 T^{2} + 277505972 T^{3} + 138199 p^{5} T^{4} + 90 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 - 1934 T + 5070371 T^{2} - 5368097468 T^{3} + 5070371 p^{5} T^{4} - 1934 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 2084 T + 8452353 T^{2} - 10305943192 T^{3} + 8452353 p^{5} T^{4} - 2084 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 - 1220 T + 13380661 T^{2} - 13300348792 T^{3} + 13380661 p^{5} T^{4} - 1220 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 - 4402 T + 39672443 T^{2} - 99874588732 T^{3} + 39672443 p^{5} T^{4} - 4402 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 + 10688 T + 96146301 T^{2} + 612569968000 T^{3} + 96146301 p^{5} T^{4} + 10688 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 8190 T + 170949451 T^{2} + 970713759316 T^{3} + 170949451 p^{5} T^{4} + 8190 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 5974 T + 299475567 T^{2} - 1256639577796 T^{3} + 299475567 p^{5} T^{4} - 5974 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 18868 T + 130667157 T^{2} + 125017085992 T^{3} + 130667157 p^{5} T^{4} - 18868 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 55500 T + 1674509341 T^{2} - 31033835957928 T^{3} + 1674509341 p^{5} T^{4} - 55500 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 - 9206 T + 983431051 T^{2} - 8550502005892 T^{3} + 983431051 p^{5} T^{4} - 9206 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 + 59196 T + 3288274801 T^{2} + 91826674965992 T^{3} + 3288274801 p^{5} T^{4} + 59196 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 79902 T + 4181653075 T^{2} - 137963205899380 T^{3} + 4181653075 p^{5} T^{4} - 79902 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 - 4468 T + 2713884569 T^{2} - 18497379943480 T^{3} + 2713884569 p^{5} T^{4} - 4468 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 + 75164 T + 5685876085 T^{2} + 240054320850568 T^{3} + 5685876085 p^{5} T^{4} + 75164 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 + 61290 T + 3793555675 T^{2} + 101810291156612 T^{3} + 3793555675 p^{5} T^{4} + 61290 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 + 83564 T + 5767840805 T^{2} + 230557669308584 T^{3} + 5767840805 p^{5} T^{4} + 83564 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 - 74764 T + 9023557773 T^{2} - 388285988035672 T^{3} + 9023557773 p^{5} T^{4} - 74764 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 37342 T + 9594205287 T^{2} - 76517859977828 T^{3} + 9594205287 p^{5} T^{4} - 37342 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 - 33486 T + 20157600991 T^{2} - 669005112374372 T^{3} + 20157600991 p^{5} T^{4} - 33486 p^{10} T^{5} + p^{15} T^{6} \)
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

−10.73589125896246282717366652844, −10.58893377518855998047145526780, −9.941995098414507627497583867431, −9.852615066011022509109132191102, −9.073521128793540909720294743676, −8.872857548402799945333864394071, −8.656401738345292073450692192141, −7.71247386472338710809304547502, −7.54564376253530081571860023514, −7.24416790679591350418546266865, −7.23804348541416272608671683331, −6.27019943724201140548148196582, −6.26807381796681273112661186578, −5.43362080904759813289245081167, −5.37706629588292453849612882878, −5.20160616164181060195187458865, −4.31461018237420543765089994123, −4.31178835262205792821061783099, −4.07827055618879432479428933903, −3.18580952412537033892790519167, −3.17794485127400430122486985968, −2.00690265717859599537468374719, −1.34916322237558547301002581774, −0.67413957804236615491802169391, −0.63533068104857048199077789612, 0.63533068104857048199077789612, 0.67413957804236615491802169391, 1.34916322237558547301002581774, 2.00690265717859599537468374719, 3.17794485127400430122486985968, 3.18580952412537033892790519167, 4.07827055618879432479428933903, 4.31178835262205792821061783099, 4.31461018237420543765089994123, 5.20160616164181060195187458865, 5.37706629588292453849612882878, 5.43362080904759813289245081167, 6.26807381796681273112661186578, 6.27019943724201140548148196582, 7.23804348541416272608671683331, 7.24416790679591350418546266865, 7.54564376253530081571860023514, 7.71247386472338710809304547502, 8.656401738345292073450692192141, 8.872857548402799945333864394071, 9.073521128793540909720294743676, 9.852615066011022509109132191102, 9.941995098414507627497583867431, 10.58893377518855998047145526780, 10.73589125896246282717366652844

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