Properties

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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 27·3-s − 56·4-s + 75·5-s − 54·6-s + 152·7-s − 144·8-s + 486·9-s + 150·10-s − 363·11-s + 1.51e3·12-s − 546·13-s + 304·14-s − 2.02e3·15-s + 1.32e3·16-s − 314·17-s + 972·18-s + 1.80e3·19-s − 4.20e3·20-s − 4.10e3·21-s − 726·22-s + 4.28e3·23-s + 3.88e3·24-s + 3.75e3·25-s − 1.09e3·26-s − 7.29e3·27-s − 8.51e3·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.73·3-s − 7/4·4-s + 1.34·5-s − 0.612·6-s + 1.17·7-s − 0.795·8-s + 2·9-s + 0.474·10-s − 0.904·11-s + 3.03·12-s − 0.896·13-s + 0.414·14-s − 2.32·15-s + 1.29·16-s − 0.263·17-s + 0.707·18-s + 1.14·19-s − 2.34·20-s − 2.03·21-s − 0.319·22-s + 1.69·23-s + 1.37·24-s + 6/5·25-s − 0.316·26-s − 1.92·27-s − 2.05·28-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

\(L(3)\) \(\approx\) \(2.875942013\)
\(L(\frac12)\) \(\approx\) \(2.875942013\)
\(L(\frac{7}{2})\) 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$S_4\times C_2$ \( 1 - p T + 15 p^{2} T^{2} - 11 p^{3} T^{3} + 15 p^{7} T^{4} - p^{11} T^{5} + p^{15} T^{6} \)
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$S_4\times C_2$ \( 1 + 103242 T + 15506277199 T^{2} + 1521607808779276 T^{3} + 15506277199 p^{5} T^{4} + 103242 p^{10} T^{5} + p^{15} T^{6} \)
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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

−10.74842132224563290986945716651, −10.16302585159666504076736976046, −9.972486242375844595378535073880, −9.638418718604824914349084907357, −9.334602781076864785980735753349, −8.851811704478158946884983000373, −8.830046302966854482740088656169, −7.994437793409911235673916520874, −7.66609064906276933676167276478, −7.54031646514276314576006592346, −6.85529341403178437414155954229, −6.39119191606861073606589229219, −6.11097717284170366577635046256, −5.37990289158597660332594887223, −5.35296769742148392019876554355, −5.26840326735560715748487049569, −4.48073631061681286237257232700, −4.41737165682017092840881371958, −4.31994535398646703916880005376, −2.93058766300651227263226916171, −2.77608166455904849518776534731, −2.03549059882497724123099092593, −1.13250124010902434281094106825, −0.77937066771802246596478905562, −0.59035267211140903457833964202, 0.59035267211140903457833964202, 0.77937066771802246596478905562, 1.13250124010902434281094106825, 2.03549059882497724123099092593, 2.77608166455904849518776534731, 2.93058766300651227263226916171, 4.31994535398646703916880005376, 4.41737165682017092840881371958, 4.48073631061681286237257232700, 5.26840326735560715748487049569, 5.35296769742148392019876554355, 5.37990289158597660332594887223, 6.11097717284170366577635046256, 6.39119191606861073606589229219, 6.85529341403178437414155954229, 7.54031646514276314576006592346, 7.66609064906276933676167276478, 7.994437793409911235673916520874, 8.830046302966854482740088656169, 8.851811704478158946884983000373, 9.334602781076864785980735753349, 9.638418718604824914349084907357, 9.972486242375844595378535073880, 10.16302585159666504076736976046, 10.74842132224563290986945716651

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