Properties

Label 6-1080e3-1.1-c3e3-0-6
Degree $6$
Conductor $1259712000$
Sign $-1$
Analytic cond. $258743.$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 15·5-s − 8·7-s + 10·11-s + 48·13-s − 37·17-s + 29·19-s + 11·23-s + 150·25-s − 28·29-s + 41·31-s + 120·35-s + 230·37-s − 370·41-s − 130·43-s + 56·47-s − 209·49-s − 805·53-s − 150·55-s − 576·59-s − 257·61-s − 720·65-s − 14·67-s − 1.23e3·71-s − 398·73-s − 80·77-s − 321·79-s − 687·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.431·7-s + 0.274·11-s + 1.02·13-s − 0.527·17-s + 0.350·19-s + 0.0997·23-s + 6/5·25-s − 0.179·29-s + 0.237·31-s + 0.579·35-s + 1.02·37-s − 1.40·41-s − 0.461·43-s + 0.173·47-s − 0.609·49-s − 2.08·53-s − 0.367·55-s − 1.27·59-s − 0.539·61-s − 1.37·65-s − 0.0255·67-s − 2.06·71-s − 0.638·73-s − 0.118·77-s − 0.457·79-s − 0.908·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(258743.\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + p T )^{3} \)
good7$S_4\times C_2$ \( 1 + 8 T + 39 p T^{2} + 11320 T^{3} + 39 p^{4} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 10 T + 1129 T^{2} + 2980 p T^{3} + 1129 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 48 T + 3891 T^{2} - 92328 T^{3} + 3891 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 37 T + 14418 T^{2} + 347965 T^{3} + 14418 p^{3} T^{4} + 37 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 29 T + 17960 T^{2} - 420497 T^{3} + 17960 p^{3} T^{4} - 29 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 11 T + 30456 T^{2} - 222899 T^{3} + 30456 p^{3} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 28 T + 55099 T^{2} + 1662544 T^{3} + 55099 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 41 T + 78964 T^{2} - 2734453 T^{3} + 78964 p^{3} T^{4} - 41 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 230 T + 114787 T^{2} - 19298596 T^{3} + 114787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 370 T + 214275 T^{2} + 48796108 T^{3} + 214275 p^{3} T^{4} + 370 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 130 T + 44673 T^{2} + 3936068 T^{3} + 44673 p^{3} T^{4} + 130 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 56 T + 115517 T^{2} - 4195536 T^{3} + 115517 p^{3} T^{4} - 56 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 805 T + 594702 T^{2} + 247415005 T^{3} + 594702 p^{3} T^{4} + 805 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 576 T + 3639 p T^{2} + 9306072 T^{3} + 3639 p^{4} T^{4} + 576 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 257 T + 202474 T^{2} + 153986701 T^{3} + 202474 p^{3} T^{4} + 257 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 14 T + 651149 T^{2} - 22117636 T^{3} + 651149 p^{3} T^{4} + 14 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 1238 T + 1448677 T^{2} + 895305068 T^{3} + 1448677 p^{3} T^{4} + 1238 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 398 T + 1211307 T^{2} + 310732420 T^{3} + 1211307 p^{3} T^{4} + 398 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 321 T + 774636 T^{2} + 459949637 T^{3} + 774636 p^{3} T^{4} + 321 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 687 T + 1037148 T^{2} + 349696983 T^{3} + 1037148 p^{3} T^{4} + 687 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 2358 T + 3793395 T^{2} + 3699365060 T^{3} + 3793395 p^{3} T^{4} + 2358 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 576 T + 2295459 T^{2} - 793944704 T^{3} + 2295459 p^{3} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.920997719762445011616983049624, −8.399009704750417456453347680403, −8.244454219238424877829273025447, −8.123223048880041412392492650052, −7.66883830973207514300053793824, −7.43053671639416948890301419574, −7.28178947492734349764003880219, −6.66623303075969103707152799338, −6.62993660017870673966316052575, −6.41815218180490151648113305357, −6.02460986526433133976582725155, −5.64878640367013704726108530694, −5.42249356570447592576916578410, −4.87687871983544592852167946789, −4.55340333393359060492386660907, −4.53529842478448284118445245390, −3.94250200854100507966225688440, −3.72269555153788799733199914875, −3.51307316574169559613093094158, −3.02273701545353796065446925274, −2.65916665768805068549757905938, −2.58239635113802501202783668781, −1.43565528541890746349950462974, −1.41329647326403708474829547054, −1.23814240489660256640411658730, 0, 0, 0, 1.23814240489660256640411658730, 1.41329647326403708474829547054, 1.43565528541890746349950462974, 2.58239635113802501202783668781, 2.65916665768805068549757905938, 3.02273701545353796065446925274, 3.51307316574169559613093094158, 3.72269555153788799733199914875, 3.94250200854100507966225688440, 4.53529842478448284118445245390, 4.55340333393359060492386660907, 4.87687871983544592852167946789, 5.42249356570447592576916578410, 5.64878640367013704726108530694, 6.02460986526433133976582725155, 6.41815218180490151648113305357, 6.62993660017870673966316052575, 6.66623303075969103707152799338, 7.28178947492734349764003880219, 7.43053671639416948890301419574, 7.66883830973207514300053793824, 8.123223048880041412392492650052, 8.244454219238424877829273025447, 8.399009704750417456453347680403, 8.920997719762445011616983049624

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