Properties

Label 6-7800e3-1.1-c1e3-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s − 3·13-s − 6·17-s + 3·19-s − 10·27-s + 3·29-s − 9·37-s + 9·39-s + 3·41-s + 6·43-s − 3·47-s − 6·49-s + 18·51-s − 3·53-s − 9·57-s + 6·59-s − 6·61-s + 3·67-s + 9·71-s + 12·73-s + 15·79-s + 15·81-s + 6·83-s − 9·87-s + 6·89-s − 12·97-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 0.832·13-s − 1.45·17-s + 0.688·19-s − 1.92·27-s + 0.557·29-s − 1.47·37-s + 1.44·39-s + 0.468·41-s + 0.914·43-s − 0.437·47-s − 6/7·49-s + 2.52·51-s − 0.412·53-s − 1.19·57-s + 0.781·59-s − 0.768·61-s + 0.366·67-s + 1.06·71-s + 1.40·73-s + 1.68·79-s + 5/3·81-s + 0.658·83-s − 0.964·87-s + 0.635·89-s − 1.21·97-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: induced by $\chi_{7800} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(2.268138295\)
\(L(\frac12)\) \(\approx\) \(2.268138295\)
\(L(\frac{3}{2})\) 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$D_{6}$ \( 1 + 6 T^{2} - 20 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 18 T^{2} + 20 T^{3} + 18 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 6 T + 48 T^{2} + 192 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 3 T + 45 T^{2} - 110 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 9 T^{2} + 80 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 + 78 T^{2} + 20 T^{3} + 78 p T^{4} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 9 T + 123 T^{2} + 638 T^{3} + 123 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 3 T + 111 T^{2} - 222 T^{3} + 111 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
47$S_4\times C_2$ \( 1 + 3 T + 114 T^{2} + 313 T^{3} + 114 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 3 T + 102 T^{2} + 179 T^{3} + 102 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 6 T + 114 T^{2} - 766 T^{3} + 114 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 60 T^{2} + 200 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 3 T + 174 T^{2} - 353 T^{3} + 174 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 9 T + 225 T^{2} - 1250 T^{3} + 225 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 12 T + 207 T^{2} - 1736 T^{3} + 207 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 15 T + 3 p T^{2} - 2270 T^{3} + 3 p^{2} T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 186 T^{2} - 654 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 219 T^{2} - 876 T^{3} + 219 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 12 T + 279 T^{2} + 2072 T^{3} + 279 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.81566403786839302717289275488, −6.64609209375297943786613701401, −6.44114388191899731309854616889, −6.42926703324054260830791983073, −5.90468482326890288913211060871, −5.86629355171247544091250758915, −5.52725678080984919635680141102, −5.07637454836518098421866529986, −5.07015838566720397000186970651, −5.05348836243339910270376436089, −4.62521586558471810135190829579, −4.35135973756512432509162840544, −4.29993548643585845828494476035, −3.66033200964660701696006502701, −3.62777619991229592778537906290, −3.49140711690956701193475700636, −2.83550251955844139055387830888, −2.64454088268343094660779094739, −2.45170719434205755002081763361, −1.80965145413848804474142019855, −1.74600218016868689569647449956, −1.62568322372382711711383602411, −0.66631521545458343069114498580, −0.58900573396654181208275420676, −0.52094666811292158527118134641, 0.52094666811292158527118134641, 0.58900573396654181208275420676, 0.66631521545458343069114498580, 1.62568322372382711711383602411, 1.74600218016868689569647449956, 1.80965145413848804474142019855, 2.45170719434205755002081763361, 2.64454088268343094660779094739, 2.83550251955844139055387830888, 3.49140711690956701193475700636, 3.62777619991229592778537906290, 3.66033200964660701696006502701, 4.29993548643585845828494476035, 4.35135973756512432509162840544, 4.62521586558471810135190829579, 5.05348836243339910270376436089, 5.07015838566720397000186970651, 5.07637454836518098421866529986, 5.52725678080984919635680141102, 5.86629355171247544091250758915, 5.90468482326890288913211060871, 6.42926703324054260830791983073, 6.44114388191899731309854616889, 6.64609209375297943786613701401, 6.81566403786839302717289275488

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