Properties

Label 6-1400e3-1.1-c1e3-0-0
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

\(L(1)\) \(\approx\) \(4.026702163\)
\(L(\frac12)\) \(\approx\) \(4.026702163\)
\(L(\frac{3}{2})\) 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

−8.669383570223460816829694740894, −8.282315010477802170753222703561, −7.956467123206446196136747544306, −7.71845745435832364682029767468, −7.35235238807244672391641168005, −7.08444132262771253789531127816, −6.72984832126464717708769967367, −6.62310445893279308196615686653, −6.42686629260159248091551996064, −6.22969908981643172555909007877, −5.59342515765788872900937549542, −5.26522351164895423757386557354, −5.22010502321518347905447888820, −4.88323502689938343491147495418, −4.30087607381132231659988055947, −4.16965028885278454720126471453, −3.61680947352390500537034183457, −3.45591462771770814431260870441, −3.28623503642047064414362320420, −2.62002873284730933572755918240, −2.40096966435580942129278789581, −2.38814431097515511130754157654, −1.38865274016735659785868412142, −0.921380942318043290863832812987, −0.66632876187762090243030286651, 0.66632876187762090243030286651, 0.921380942318043290863832812987, 1.38865274016735659785868412142, 2.38814431097515511130754157654, 2.40096966435580942129278789581, 2.62002873284730933572755918240, 3.28623503642047064414362320420, 3.45591462771770814431260870441, 3.61680947352390500537034183457, 4.16965028885278454720126471453, 4.30087607381132231659988055947, 4.88323502689938343491147495418, 5.22010502321518347905447888820, 5.26522351164895423757386557354, 5.59342515765788872900937549542, 6.22969908981643172555909007877, 6.42686629260159248091551996064, 6.62310445893279308196615686653, 6.72984832126464717708769967367, 7.08444132262771253789531127816, 7.35235238807244672391641168005, 7.71845745435832364682029767468, 7.956467123206446196136747544306, 8.282315010477802170753222703561, 8.669383570223460816829694740894

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