Properties

Label 6-99e3-1.1-c5e3-0-0
Degree $6$
Conductor $970299$
Sign $1$
Analytic cond. $4003.01$
Root an. cond. $3.98472$
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  − 6·4-s − 24·5-s + 84·7-s + 188·8-s − 363·11-s + 486·13-s − 156·16-s − 1.08e3·17-s + 1.38e3·19-s + 144·20-s + 3.06e3·23-s − 4.42e3·25-s − 504·28-s + 3.42e3·29-s − 4.09e3·31-s − 2.25e3·32-s − 2.01e3·35-s + 1.77e4·37-s − 4.51e3·40-s − 5.99e3·41-s − 2.62e4·43-s + 2.17e3·44-s + 1.72e4·47-s + 2.58e3·49-s − 2.91e3·52-s − 5.05e4·53-s + 8.71e3·55-s + ⋯
L(s)  = 1  − 0.187·4-s − 0.429·5-s + 0.647·7-s + 1.03·8-s − 0.904·11-s + 0.797·13-s − 0.152·16-s − 0.911·17-s + 0.876·19-s + 0.0804·20-s + 1.20·23-s − 1.41·25-s − 0.121·28-s + 0.756·29-s − 0.765·31-s − 0.389·32-s − 0.278·35-s + 2.12·37-s − 0.445·40-s − 0.556·41-s − 2.16·43-s + 0.169·44-s + 1.13·47-s + 0.153·49-s − 0.149·52-s − 2.47·53-s + 0.388·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 970299 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 970299 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(970299\)    =    \(3^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(4003.01\)
Root analytic conductor: \(3.98472\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 970299,\ (\ :5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.741405628\)
\(L(\frac12)\) \(\approx\) \(2.741405628\)
\(L(\frac{7}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + p^{2} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 3 p T^{2} - 47 p^{2} T^{3} + 3 p^{6} T^{4} + p^{15} T^{6} \)
5$S_4\times C_2$ \( 1 + 24 T + 5004 T^{2} + 111046 T^{3} + 5004 p^{5} T^{4} + 24 p^{10} T^{5} + p^{15} T^{6} \)
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

−11.46223527359376072936181711535, −11.11788636040633740158406670087, −10.85816404567449948877611221452, −10.61016534916104707453940854104, −10.13081932323675718413416094089, −9.611973719372076875384510614881, −9.425850760959062506184125156646, −8.771647902938532080187999640549, −8.557977594283157908639199609071, −8.020059094408573918788279194839, −7.66615299119126929011214717355, −7.55480085578745902083868749668, −7.04390767690774017755568924249, −6.33445668466258336422048453654, −6.14638485831882344594985707392, −5.43546598703918022542606559794, −4.80346167995890821190125612942, −4.78997703276040416983981787911, −4.29002433717049295604620439118, −3.36482965898573701239353172915, −3.33925794033852711775931988448, −2.22068982757566119899778413809, −1.86065374731957022472075225018, −1.07359678769425008406181978019, −0.46102992735699154315282966430, 0.46102992735699154315282966430, 1.07359678769425008406181978019, 1.86065374731957022472075225018, 2.22068982757566119899778413809, 3.33925794033852711775931988448, 3.36482965898573701239353172915, 4.29002433717049295604620439118, 4.78997703276040416983981787911, 4.80346167995890821190125612942, 5.43546598703918022542606559794, 6.14638485831882344594985707392, 6.33445668466258336422048453654, 7.04390767690774017755568924249, 7.55480085578745902083868749668, 7.66615299119126929011214717355, 8.020059094408573918788279194839, 8.557977594283157908639199609071, 8.771647902938532080187999640549, 9.425850760959062506184125156646, 9.611973719372076875384510614881, 10.13081932323675718413416094089, 10.61016534916104707453940854104, 10.85816404567449948877611221452, 11.11788636040633740158406670087, 11.46223527359376072936181711535

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