Properties

Label 6-4680e3-1.1-c1e3-0-0
Degree $6$
Conductor $102503232000$
Sign $1$
Analytic cond. $52187.7$
Root an. cond. $6.11309$
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  − 3·5-s + 7-s − 5·11-s − 3·13-s − 7·17-s + 6·19-s + 23-s + 6·25-s − 10·29-s + 2·31-s − 3·35-s + 11·37-s + 41-s + 10·47-s − 3·53-s + 15·55-s − 8·59-s − 3·61-s + 9·65-s − 12·67-s + 19·71-s + 26·73-s − 5·77-s + 19·79-s + 4·83-s + 21·85-s − 13·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s − 1.50·11-s − 0.832·13-s − 1.69·17-s + 1.37·19-s + 0.208·23-s + 6/5·25-s − 1.85·29-s + 0.359·31-s − 0.507·35-s + 1.80·37-s + 0.156·41-s + 1.45·47-s − 0.412·53-s + 2.02·55-s − 1.04·59-s − 0.384·61-s + 1.11·65-s − 1.46·67-s + 2.25·71-s + 3.04·73-s − 0.569·77-s + 2.13·79-s + 0.439·83-s + 2.27·85-s − 1.37·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(52187.7\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.546008797\)
\(L(\frac12)\) \(\approx\) \(1.546008797\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
13$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - T + T^{2} + 26 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 5 T + 25 T^{2} + 78 T^{3} + 25 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 7 T - T^{2} - 118 T^{3} - p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 6 T + 41 T^{2} - 212 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 49 T^{2} - 6 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 75 T^{2} + 396 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 29 T^{2} - 156 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 11 T + 135 T^{2} - 794 T^{3} + 135 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - T + 107 T^{2} - 86 T^{3} + 107 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 17 T^{2} + 256 T^{3} + 17 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 10 T + 109 T^{2} - 684 T^{3} + 109 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 3 T + 47 T^{2} + 154 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 17 T^{2} - 80 T^{3} + 17 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 3 T + 71 T^{2} + 202 T^{3} + 71 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 12 T + 137 T^{2} + 968 T^{3} + 137 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 19 T + 317 T^{2} - 2858 T^{3} + 317 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 26 T + 399 T^{2} - 4124 T^{3} + 399 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 19 T + 341 T^{2} - 3162 T^{3} + 341 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 73 T^{2} - 984 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 13 T + 307 T^{2} + 2334 T^{3} + 307 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 9 T + 135 T^{2} - 326 T^{3} + 135 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.45059995352314743278744328710, −7.29601248160645284847637466286, −6.89977806415916137912556072151, −6.71540403167338805470065555144, −6.40890626933003294616995461830, −6.13484908617887122548345975936, −5.87287347640101518731864392510, −5.57686208155334947777720405417, −5.18071342372119117624144933659, −5.03452532530660049121570644069, −4.93086978502295908104881366942, −4.66664519377771214813895867618, −4.30702375748260935994724650855, −3.96871042707624413497712826668, −3.87369940944715429293985538171, −3.58124676219254280632695343550, −3.03140229178168389599854464624, −2.83528862120194662445192562441, −2.80397745882173519089030859073, −2.19200091241778254745470924294, −1.89729809055040048663653350488, −1.89243048336532050898243701258, −0.836348112622736929936553495723, −0.76194580856367103969879017422, −0.32002839491010516913047561627, 0.32002839491010516913047561627, 0.76194580856367103969879017422, 0.836348112622736929936553495723, 1.89243048336532050898243701258, 1.89729809055040048663653350488, 2.19200091241778254745470924294, 2.80397745882173519089030859073, 2.83528862120194662445192562441, 3.03140229178168389599854464624, 3.58124676219254280632695343550, 3.87369940944715429293985538171, 3.96871042707624413497712826668, 4.30702375748260935994724650855, 4.66664519377771214813895867618, 4.93086978502295908104881366942, 5.03452532530660049121570644069, 5.18071342372119117624144933659, 5.57686208155334947777720405417, 5.87287347640101518731864392510, 6.13484908617887122548345975936, 6.40890626933003294616995461830, 6.71540403167338805470065555144, 6.89977806415916137912556072151, 7.29601248160645284847637466286, 7.45059995352314743278744328710

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