Properties

Label 6-1380e3-1.1-c1e3-0-0
Degree $6$
Conductor $2628072000$
Sign $1$
Analytic cond. $1338.03$
Root an. cond. $3.31954$
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  + 3·3-s − 3·5-s + 2·7-s + 6·9-s + 4·11-s + 2·13-s − 9·15-s + 10·19-s + 6·21-s + 3·23-s + 6·25-s + 10·27-s + 10·31-s + 12·33-s − 6·35-s + 2·37-s + 6·39-s + 8·41-s + 16·43-s − 18·45-s − 4·47-s − 2·49-s + 8·53-s − 12·55-s + 30·57-s + 10·61-s + 12·63-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s + 1.20·11-s + 0.554·13-s − 2.32·15-s + 2.29·19-s + 1.30·21-s + 0.625·23-s + 6/5·25-s + 1.92·27-s + 1.79·31-s + 2.08·33-s − 1.01·35-s + 0.328·37-s + 0.960·39-s + 1.24·41-s + 2.43·43-s − 2.68·45-s − 0.583·47-s − 2/7·49-s + 1.09·53-s − 1.61·55-s + 3.97·57-s + 1.28·61-s + 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(1338.03\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1380} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.847312962\)
\(L(\frac12)\) \(\approx\) \(8.847312962\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
23$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 - 2 T + 6 T^{2} - 4 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 4 T + 19 T^{2} - 40 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 2 T + 21 T^{2} - 64 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 30 T^{2} + 18 T^{3} + 30 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 10 T + 71 T^{2} - 328 T^{3} + 71 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 66 T^{2} - 18 T^{3} + 66 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 10 T + 110 T^{2} - 616 T^{3} + 110 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T + 48 T^{2} - 256 T^{3} + 48 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 8 T + 22 T^{2} - 74 T^{3} + 22 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 16 T + 149 T^{2} - 1072 T^{3} + 149 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 91 T^{2} + 160 T^{3} + 91 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 8 T + 58 T^{2} - 266 T^{3} + 58 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 156 T^{2} + 18 T^{3} + 156 p T^{4} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 10 T + 161 T^{2} - 928 T^{3} + 161 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 10 T + 218 T^{2} - 1336 T^{3} + 218 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 8 T + 112 T^{2} + 554 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
73$D_{6}$ \( 1 - 18 T + 153 T^{2} - 1136 T^{3} + 153 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 2116 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 10 T + 232 T^{2} - 1432 T^{3} + 232 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 67 T^{2} + 556 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 20 T + 203 T^{2} + 1544 T^{3} + 203 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.586356637424984742981632266419, −7.989372098118033589802635454201, −7.973207165389061169146942735373, −7.953559415058402442644368898593, −7.48946655346227699858853883078, −7.36106031408583522974286330676, −7.04304329945705091762746347712, −6.57819038218311113684043615644, −6.46511903164147711739279338069, −6.25210672291659146773954580449, −5.43287411975762953409106091050, −5.39746182357178917946156614273, −5.10597839717954388070057587370, −4.51317189637090511227030872574, −4.33704916549521674567642155819, −4.19296708160366480358094398543, −3.56670292035907263607382784903, −3.49862922096268542721387489359, −3.45889390362862916763654605637, −2.65190881754065178858108018149, −2.47009396817158378024093374842, −2.29904300338527129077219014903, −1.22767801999255964670496877028, −1.10076524099316238344841236551, −0.960568129185171567134975236221, 0.960568129185171567134975236221, 1.10076524099316238344841236551, 1.22767801999255964670496877028, 2.29904300338527129077219014903, 2.47009396817158378024093374842, 2.65190881754065178858108018149, 3.45889390362862916763654605637, 3.49862922096268542721387489359, 3.56670292035907263607382784903, 4.19296708160366480358094398543, 4.33704916549521674567642155819, 4.51317189637090511227030872574, 5.10597839717954388070057587370, 5.39746182357178917946156614273, 5.43287411975762953409106091050, 6.25210672291659146773954580449, 6.46511903164147711739279338069, 6.57819038218311113684043615644, 7.04304329945705091762746347712, 7.36106031408583522974286330676, 7.48946655346227699858853883078, 7.953559415058402442644368898593, 7.973207165389061169146942735373, 7.989372098118033589802635454201, 8.586356637424984742981632266419

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