Properties

Label 6-1400e3-1.1-c1e3-0-1
Degree $6$
Conductor $2744000000$
Sign $1$
Analytic cond. $1397.06$
Root an. cond. $3.34350$
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-s + 3·7-s − 2·9-s + 7·11-s + 5·13-s − 17-s + 4·19-s − 3·21-s − 6·23-s − 3·27-s + 3·29-s + 10·31-s − 7·33-s + 18·37-s − 5·39-s + 18·41-s − 9·47-s + 6·49-s + 51-s + 10·53-s − 4·57-s + 6·59-s + 24·61-s − 6·63-s − 10·67-s + 6·69-s + 4·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s − 2/3·9-s + 2.11·11-s + 1.38·13-s − 0.242·17-s + 0.917·19-s − 0.654·21-s − 1.25·23-s − 0.577·27-s + 0.557·29-s + 1.79·31-s − 1.21·33-s + 2.95·37-s − 0.800·39-s + 2.81·41-s − 1.31·47-s + 6/7·49-s + 0.140·51-s + 1.37·53-s − 0.529·57-s + 0.781·59-s + 3.07·61-s − 0.755·63-s − 1.22·67-s + 0.722·69-s + 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{6} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(1397.06\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.501989881\)
\(L(\frac12)\) \(\approx\) \(4.501989881\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 7 T + 41 T^{2} - 146 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 5 T + 17 T^{2} - 24 T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + T + 27 T^{2} + 54 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 4 T + 43 T^{2} - 160 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 10 T + 101 T^{2} - 540 T^{3} + 101 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
41$S_4\times C_2$ \( 1 - 18 T + 191 T^{2} - 1388 T^{3} + 191 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 89 T^{2} + 64 T^{3} + 89 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 9 T + 93 T^{2} + 614 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 115 T^{2} - 588 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 6 T + 99 T^{2} - 752 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 365 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 10 T + 137 T^{2} + 828 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 6 T + 191 T^{2} + 740 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 17 T + 205 T^{2} - 2138 T^{3} + 205 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 107 T^{2} + 1168 T^{3} + 107 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 95 T^{2} - 464 T^{3} + 95 p T^{4} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 9 T + 275 T^{2} + 1702 T^{3} + 275 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.648048244258167733590530724345, −8.153153746814121881136114579261, −7.930551893332694010655160734481, −7.81986938989125919202368635392, −7.53382253460765391874189245052, −7.05424072649334404947060653047, −6.76489926816160492191777470339, −6.39766076663441368803609041224, −6.19651028174567977770836575672, −6.16163814420835991870453845675, −5.63410519511835109250310457283, −5.52228032222153400636192766502, −5.30534279764248165983535326127, −4.54379647525295333219030650748, −4.41758675073260367377563886519, −4.20529324312616693455694933234, −4.04269013451957889427795671226, −3.42752006728646510429893134776, −3.38255358477665102765446509456, −2.49697112735073636745183363020, −2.47191537705500274173708198819, −1.99030140242869426828584440674, −1.23459275826876975863540179242, −1.03381490661590609565760309995, −0.77529960755431526406223353922, 0.77529960755431526406223353922, 1.03381490661590609565760309995, 1.23459275826876975863540179242, 1.99030140242869426828584440674, 2.47191537705500274173708198819, 2.49697112735073636745183363020, 3.38255358477665102765446509456, 3.42752006728646510429893134776, 4.04269013451957889427795671226, 4.20529324312616693455694933234, 4.41758675073260367377563886519, 4.54379647525295333219030650748, 5.30534279764248165983535326127, 5.52228032222153400636192766502, 5.63410519511835109250310457283, 6.16163814420835991870453845675, 6.19651028174567977770836575672, 6.39766076663441368803609041224, 6.76489926816160492191777470339, 7.05424072649334404947060653047, 7.53382253460765391874189245052, 7.81986938989125919202368635392, 7.930551893332694010655160734481, 8.153153746814121881136114579261, 8.648048244258167733590530724345

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