Properties

Label 6-275e3-1.1-c5e3-0-0
Degree $6$
Conductor $20796875$
Sign $1$
Analytic cond. $85798.5$
Root an. cond. $6.64120$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 34·3-s − 6·4-s − 84·7-s + 188·8-s + 210·9-s + 363·11-s + 204·12-s − 486·13-s − 156·16-s − 1.08e3·17-s + 1.38e3·19-s + 2.85e3·21-s + 3.06e3·23-s − 6.39e3·24-s + 9.71e3·27-s + 504·28-s − 3.42e3·29-s − 4.09e3·31-s − 2.25e3·32-s − 1.23e4·33-s − 1.26e3·36-s − 1.77e4·37-s + 1.65e4·39-s + 5.99e3·41-s + 2.62e4·43-s − 2.17e3·44-s + 1.72e4·47-s + ⋯
L(s)  = 1  − 2.18·3-s − 0.187·4-s − 0.647·7-s + 1.03·8-s + 0.864·9-s + 0.904·11-s + 0.408·12-s − 0.797·13-s − 0.152·16-s − 0.911·17-s + 0.876·19-s + 1.41·21-s + 1.20·23-s − 2.26·24-s + 2.56·27-s + 0.121·28-s − 0.756·29-s − 0.765·31-s − 0.389·32-s − 1.97·33-s − 0.162·36-s − 2.12·37-s + 1.73·39-s + 0.556·41-s + 2.16·43-s − 0.169·44-s + 1.13·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20796875 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20796875 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(20796875\)    =    \(5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(85798.5\)
Root analytic conductor: \(6.64120\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 20796875,\ (\ :5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.120325212\)
\(L(\frac12)\) \(\approx\) \(1.120325212\)
\(L(\frac{7}{2})\) 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)$
bad5 \( 1 \)
11$C_1$ \( ( 1 - p^{2} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 3 p T^{2} - 47 p^{2} T^{3} + 3 p^{6} T^{4} + p^{15} T^{6} \)
3$S_4\times C_2$ \( 1 + 34 T + 946 T^{2} + 5104 p T^{3} + 946 p^{5} T^{4} + 34 p^{10} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 + 12 p T + 639 p T^{2} - 2556872 T^{3} + 639 p^{6} T^{4} + 12 p^{11} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 + 486 T + 767415 T^{2} + 196760188 T^{3} + 767415 p^{5} T^{4} + 486 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 + 1086 T + 2690223 T^{2} + 2752177348 T^{3} + 2690223 p^{5} T^{4} + 1086 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 1380 T + 7644297 T^{2} - 6777009240 T^{3} + 7644297 p^{5} T^{4} - 1380 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 - 3066 T + 8993526 T^{2} - 22463329348 T^{3} + 8993526 p^{5} T^{4} - 3066 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 + 3426 T + 62121159 T^{2} + 136513203828 T^{3} + 62121159 p^{5} T^{4} + 3426 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 + 4098 T + 90136878 T^{2} + 235738865996 T^{3} + 90136878 p^{5} T^{4} + 4098 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 17724 T + 229846956 T^{2} + 1916316420702 T^{3} + 229846956 p^{5} T^{4} + 17724 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 5994 T + 174379803 T^{2} - 1186954316020 T^{3} + 174379803 p^{5} T^{4} - 5994 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 26208 T + 443706117 T^{2} - 5261719449744 T^{3} + 443706117 p^{5} T^{4} - 26208 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 17232 T + 689784333 T^{2} - 7833971382112 T^{3} + 689784333 p^{5} T^{4} - 17232 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 + 50586 T + 1969881291 T^{2} + 44160585727452 T^{3} + 1969881291 p^{5} T^{4} + 50586 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 + 3738 T + 1293303186 T^{2} + 13104411496384 T^{3} + 1293303186 p^{5} T^{4} + 3738 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 18486 T + 1754869767 T^{2} - 15992539689564 T^{3} + 1754869767 p^{5} T^{4} - 18486 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 - 47754 T + 997454514 T^{2} + 18340812610856 T^{3} + 997454514 p^{5} T^{4} - 47754 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 39282 T + 4433022990 T^{2} - 143037873283668 T^{3} + 4433022990 p^{5} T^{4} - 39282 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 + 15426 T + 2562304635 T^{2} + 98498106053188 T^{3} + 2562304635 p^{5} T^{4} + 15426 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 - 125148 T + 13122635793 T^{2} - 768895025227784 T^{3} + 13122635793 p^{5} T^{4} - 125148 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 - 143928 T + 12127124157 T^{2} - 722278658611584 T^{3} + 12127124157 p^{5} T^{4} - 143928 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 + 106824 T + 17674467768 T^{2} + 1102702152985302 T^{3} + 17674467768 p^{5} T^{4} + 106824 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 + 9684 T + 23649435576 T^{2} + 176541508624682 T^{3} + 23649435576 p^{5} T^{4} + 9684 p^{10} T^{5} + p^{15} T^{6} \)
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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

−9.780198745835904781353694116437, −9.466695285449491543842676573575, −9.116981036543234596012167409161, −9.067187925938340861214105657075, −8.512849118021948577334567697177, −8.191318959702632640408261908619, −7.45271588410601440274222711229, −7.29335521047227507521793580751, −7.19722386673667577109874220289, −6.58754895490174052212465862238, −6.32472409781639305826337106377, −6.03700286568357450502617650611, −5.72787494427698443636528651508, −5.05685942847644630185280020666, −5.00180071673989413505652367014, −4.99549958940720481493721546621, −4.15743506606679071490815073544, −3.87408513717505939821710580020, −3.25811132264213049386467994998, −2.92987492044587280063579488857, −2.10440941980740227778365664248, −1.85941933062218914924255713192, −0.973661936740856598300973494539, −0.59739428019972024605876443052, −0.38803083625465171329291258854, 0.38803083625465171329291258854, 0.59739428019972024605876443052, 0.973661936740856598300973494539, 1.85941933062218914924255713192, 2.10440941980740227778365664248, 2.92987492044587280063579488857, 3.25811132264213049386467994998, 3.87408513717505939821710580020, 4.15743506606679071490815073544, 4.99549958940720481493721546621, 5.00180071673989413505652367014, 5.05685942847644630185280020666, 5.72787494427698443636528651508, 6.03700286568357450502617650611, 6.32472409781639305826337106377, 6.58754895490174052212465862238, 7.19722386673667577109874220289, 7.29335521047227507521793580751, 7.45271588410601440274222711229, 8.191318959702632640408261908619, 8.512849118021948577334567697177, 9.067187925938340861214105657075, 9.116981036543234596012167409161, 9.466695285449491543842676573575, 9.780198745835904781353694116437

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