Properties

Label 6-7800e3-1.1-c1e3-0-5
Degree $6$
Conductor $474552000000$
Sign $1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·7-s + 6·9-s + 2·11-s + 3·13-s + 2·17-s + 3·19-s − 6·21-s + 4·23-s − 10·27-s + 29-s + 14·31-s − 6·33-s + 37-s − 9·39-s + 5·41-s + 6·43-s − 5·47-s − 12·49-s − 6·51-s − 15·53-s − 9·57-s + 8·59-s + 18·61-s + 12·63-s + 3·67-s − 12·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.755·7-s + 2·9-s + 0.603·11-s + 0.832·13-s + 0.485·17-s + 0.688·19-s − 1.30·21-s + 0.834·23-s − 1.92·27-s + 0.185·29-s + 2.51·31-s − 1.04·33-s + 0.164·37-s − 1.44·39-s + 0.780·41-s + 0.914·43-s − 0.729·47-s − 1.71·49-s − 0.840·51-s − 2.06·53-s − 1.19·57-s + 1.04·59-s + 2.30·61-s + 1.51·63-s + 0.366·67-s − 1.44·69-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.802763592\)
\(L(\frac12)\) \(\approx\) \(4.802763592\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 - 2 T + 16 T^{2} - 20 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 4 T^{2} + 36 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 44 T^{2} - 64 T^{3} + 44 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 3 T + 17 T^{2} - 14 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - T + 54 T^{2} - 57 T^{3} + 54 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 14 T + 128 T^{2} - 768 T^{3} + 128 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - T + 87 T^{2} - 94 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 5 T + 107 T^{2} - 346 T^{3} + 107 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 6 T + 101 T^{2} - 508 T^{3} + 101 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 5 T + 130 T^{2} + 411 T^{3} + 130 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 15 T + 194 T^{2} + 1579 T^{3} + 194 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 80 T^{2} - 274 T^{3} + 80 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 18 T + 200 T^{2} - 1532 T^{3} + 200 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 3 T + 14 T^{2} - 341 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 23 T + 365 T^{2} - 3502 T^{3} + 365 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 8 T + 207 T^{2} + 1088 T^{3} + 207 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 7 T + 125 T^{2} - 822 T^{3} + 125 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 8 T + 208 T^{2} + 1158 T^{3} + 208 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 14 T + 235 T^{2} + 2364 T^{3} + 235 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 9 T^{2} + 272 T^{3} - 9 p T^{4} + p^{3} T^{6} \)
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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

−6.91287890109926689397638981891, −6.52235762616392691878129591883, −6.48656465712215944311845932937, −6.42371745310255394679206949906, −5.85785052532985411325202301245, −5.84555850954056567130427870281, −5.58505679949814194761449734491, −5.17324013302925341961322460105, −5.15778848764940520258695333935, −4.89801923147723076273300320232, −4.50360210309540243965134875292, −4.48139705919317820498532377454, −4.25420808248985440632632188487, −3.69453926412329594451076877470, −3.57445013550573118374694001865, −3.53551924642668006880083543848, −2.81491138345144566489680797618, −2.62199306089391723577718638440, −2.61678900211238879360640326458, −1.76774773119575064710442119044, −1.56821581285309629810272338961, −1.50108316716807401137140963875, −0.907818607186173803066150140910, −0.68108905407635264500246216637, −0.56446567111075060653312758195, 0.56446567111075060653312758195, 0.68108905407635264500246216637, 0.907818607186173803066150140910, 1.50108316716807401137140963875, 1.56821581285309629810272338961, 1.76774773119575064710442119044, 2.61678900211238879360640326458, 2.62199306089391723577718638440, 2.81491138345144566489680797618, 3.53551924642668006880083543848, 3.57445013550573118374694001865, 3.69453926412329594451076877470, 4.25420808248985440632632188487, 4.48139705919317820498532377454, 4.50360210309540243965134875292, 4.89801923147723076273300320232, 5.15778848764940520258695333935, 5.17324013302925341961322460105, 5.58505679949814194761449734491, 5.84555850954056567130427870281, 5.85785052532985411325202301245, 6.42371745310255394679206949906, 6.48656465712215944311845932937, 6.52235762616392691878129591883, 6.91287890109926689397638981891

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