Properties

Label 6-8550e3-1.1-c1e3-0-1
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 + 2·7-s − 10·8-s − 2·11-s − 6·13-s − 6·14-s + 15·16-s − 12·17-s + 3·19-s + 6·22-s + 2·23-s + 18·26-s + 12·28-s − 6·29-s + 8·31-s − 21·32-s + 36·34-s − 37-s − 9·38-s − 14·43-s − 12·44-s − 6·46-s − 3·47-s − 4·49-s − 36·52-s + 10·53-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s + 0.755·7-s − 3.53·8-s − 0.603·11-s − 1.66·13-s − 1.60·14-s + 15/4·16-s − 2.91·17-s + 0.688·19-s + 1.27·22-s + 0.417·23-s + 3.53·26-s + 2.26·28-s − 1.11·29-s + 1.43·31-s − 3.71·32-s + 6.17·34-s − 0.164·37-s − 1.45·38-s − 2.13·43-s − 1.80·44-s − 0.884·46-s − 0.437·47-s − 4/7·49-s − 4.99·52-s + 1.37·53-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

\(L(1)\) \(\approx\) \(0.4612954927\)
\(L(\frac12)\) \(\approx\) \(0.4612954927\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 - 2 T + 8 T^{2} - 29 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 9 T^{2} + 20 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 6 T + 36 T^{2} + 121 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 12 T + 84 T^{2} + 399 T^{3} + 84 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T + 18 T^{2} - 203 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 6 T + 84 T^{2} + 339 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 T + 89 T^{2} - 440 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + T + 86 T^{2} + 73 T^{3} + 86 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
41$C_2$ \( ( 1 + p T^{2} )^{3} \)
43$S_4\times C_2$ \( 1 + 14 T + 169 T^{2} + 1180 T^{3} + 169 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 3 T + 84 T^{2} + 327 T^{3} + 84 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 54 T^{2} - 193 T^{3} + 54 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T - 24 T^{2} - 723 T^{3} - 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 2 T + 127 T^{2} + 124 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 4 T + 178 T^{2} - 461 T^{3} + 178 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 21 T^{2} + 660 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 12 T + 192 T^{2} + 1435 T^{3} + 192 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 20 T + 269 T^{2} - 2840 T^{3} + 269 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 141 T^{2} - 832 T^{3} + 141 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 14 T + 231 T^{2} + 2180 T^{3} + 231 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 28 T + 335 T^{2} + 2992 T^{3} + 335 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.07145964362155827568576902396, −6.68085154898125034539412318525, −6.64262732052179271302170045102, −6.49546702654402310999984706128, −5.92012973995622319636226417224, −5.91989154704053771698240690137, −5.48457584163991225089452602843, −5.21584799228143375362707506645, −5.00184010162464656513853689742, −4.92578600689776400714057995082, −4.34745191904989399585305937749, −4.31759631628925177117211266726, −4.27559720964908123676558172096, −3.43173422872538939244509764568, −3.39244050325859787080489787289, −3.04932136079510884204164235486, −2.73467663422044861728572007819, −2.39928089635169478746267491379, −2.34723620830707662855278933327, −1.81744556798336408316517956451, −1.78326952862109232266535767008, −1.58090406644084436406961904211, −0.859179324259370373552832318397, −0.52944314001153072993704818918, −0.23599053022095932960090973057, 0.23599053022095932960090973057, 0.52944314001153072993704818918, 0.859179324259370373552832318397, 1.58090406644084436406961904211, 1.78326952862109232266535767008, 1.81744556798336408316517956451, 2.34723620830707662855278933327, 2.39928089635169478746267491379, 2.73467663422044861728572007819, 3.04932136079510884204164235486, 3.39244050325859787080489787289, 3.43173422872538939244509764568, 4.27559720964908123676558172096, 4.31759631628925177117211266726, 4.34745191904989399585305937749, 4.92578600689776400714057995082, 5.00184010162464656513853689742, 5.21584799228143375362707506645, 5.48457584163991225089452602843, 5.91989154704053771698240690137, 5.92012973995622319636226417224, 6.49546702654402310999984706128, 6.64262732052179271302170045102, 6.68085154898125034539412318525, 7.07145964362155827568576902396

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