Properties

Label 6-8550e3-1.1-c1e3-0-4
Degree $6$
Conductor $625026375000$
Sign $1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s + 4·7-s − 10·8-s + 8·13-s − 12·14-s + 15·16-s + 2·17-s + 3·19-s − 24·26-s + 24·28-s + 8·29-s + 4·31-s − 21·32-s − 6·34-s + 14·37-s − 9·38-s − 2·41-s + 18·43-s + 14·47-s − 4·49-s + 48·52-s − 16·53-s − 40·56-s − 24·58-s − 2·59-s − 30·61-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s + 1.51·7-s − 3.53·8-s + 2.21·13-s − 3.20·14-s + 15/4·16-s + 0.485·17-s + 0.688·19-s − 4.70·26-s + 4.53·28-s + 1.48·29-s + 0.718·31-s − 3.71·32-s − 1.02·34-s + 2.30·37-s − 1.45·38-s − 0.312·41-s + 2.74·43-s + 2.04·47-s − 4/7·49-s + 6.65·52-s − 2.19·53-s − 5.34·56-s − 3.15·58-s − 0.260·59-s − 3.84·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.888967931\)
\(L(\frac12)\) \(\approx\) \(3.888967931\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
3 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 - 4 T + 20 T^{2} - 54 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 23 T^{2} + 8 T^{3} + 23 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 8 T + 4 p T^{2} - 206 T^{3} + 4 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 44 T^{2} - 64 T^{3} + 44 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 20 T^{2} - 122 T^{3} + 20 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 8 T + 36 T^{2} - 54 T^{3} + 36 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 4 T + p T^{2} - 16 T^{3} + p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 14 T + 151 T^{2} - 1052 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 73 T^{2} + 264 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 18 T + 227 T^{2} - 1696 T^{3} + 227 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 14 T + 173 T^{2} - 1252 T^{3} + 173 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 16 T + 236 T^{2} + 34 p T^{3} + 236 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 2 T + 148 T^{2} + 156 T^{3} + 148 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 30 T + 473 T^{2} + 4552 T^{3} + 473 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 2 T + 140 T^{2} - 204 T^{3} + 140 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 91 T^{2} - 120 T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 124 T^{2} - 1624 T^{3} + 124 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T^{2} - 880 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 221 T^{2} - 988 T^{3} + 221 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 313 T^{2} - 2472 T^{3} + 313 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 10 T + 191 T^{2} - 1452 T^{3} + 191 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.04127002410583949415528556221, −6.50057272991551192611800187294, −6.45740885183728101435943110115, −6.23770355235695113068603570636, −6.06986418251652417888396084053, −5.94795916414136304136657647434, −5.75422058633682661154649791342, −5.15681601453684626527616462218, −5.07360638207371540966057529651, −4.82775545867929385239795881132, −4.42029328367395343590608225728, −4.20460900016793654116036726297, −4.18027284825562370719942424929, −3.41375804390819842166373868506, −3.39088482807366328644938631206, −3.23617361843916305963607778766, −2.76843308926410496353325140791, −2.41466119881694596788898804192, −2.37895068294696104338308469752, −1.82708751974152589583362468413, −1.50974072729124478760353690694, −1.35610572402684858222038103211, −0.921017425114854591707469467823, −0.880746390565117747533061792827, −0.49530377596380330979480927438, 0.49530377596380330979480927438, 0.880746390565117747533061792827, 0.921017425114854591707469467823, 1.35610572402684858222038103211, 1.50974072729124478760353690694, 1.82708751974152589583362468413, 2.37895068294696104338308469752, 2.41466119881694596788898804192, 2.76843308926410496353325140791, 3.23617361843916305963607778766, 3.39088482807366328644938631206, 3.41375804390819842166373868506, 4.18027284825562370719942424929, 4.20460900016793654116036726297, 4.42029328367395343590608225728, 4.82775545867929385239795881132, 5.07360638207371540966057529651, 5.15681601453684626527616462218, 5.75422058633682661154649791342, 5.94795916414136304136657647434, 6.06986418251652417888396084053, 6.23770355235695113068603570636, 6.45740885183728101435943110115, 6.50057272991551192611800187294, 7.04127002410583949415528556221

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