Properties

Label 6-58e6-4.3-c0e3-0-1
Degree $6$
Conductor $38068692544$
Sign $1$
Analytic cond. $4.73194$
Root an. cond. $1.29570$
Motivic weight $0$
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  + 3·2-s + 6·4-s − 5-s + 10·8-s + 3·9-s − 3·10-s − 13-s + 15·16-s − 17-s + 9·18-s − 6·20-s − 3·26-s + 21·32-s − 3·34-s + 18·36-s − 37-s − 10·40-s − 41-s − 3·45-s + 3·49-s − 6·52-s − 53-s − 61-s + 28·64-s + 65-s − 6·68-s + 30·72-s + ⋯
L(s)  = 1  + 3·2-s + 6·4-s − 5-s + 10·8-s + 3·9-s − 3·10-s − 13-s + 15·16-s − 17-s + 9·18-s − 6·20-s − 3·26-s + 21·32-s − 3·34-s + 18·36-s − 37-s − 10·40-s − 41-s − 3·45-s + 3·49-s − 6·52-s − 53-s − 61-s + 28·64-s + 65-s − 6·68-s + 30·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(4.73194\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3364} (1683, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{6} \cdot 29^{6} ,\ ( \ : 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(15.15638717\)
\(L(\frac12)\) \(\approx\) \(15.15638717\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{3} \)
29 \( 1 \)
good3$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
5$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
13$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
17$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
41$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
53$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
61$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
97$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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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

−7.74965213848326785022799319923, −7.25497963733971226059976244055, −7.18470734165000147642600793946, −6.99549741217795448239305223620, −6.81732356364236332115922872824, −6.61642633441396966534873934476, −6.32769406293269554289061641139, −5.96629303561939938560409251139, −5.57059668384447078051761608503, −5.53338129185782295061768046643, −5.03023761593222638131806825818, −4.83894833729736448002146758436, −4.60589431097303281457343355029, −4.35923000731639677080293535208, −4.30193486698814366703086794389, −3.98320724455297996743241071986, −3.64982058917725737759659411604, −3.50891964819255802107469484730, −3.24311040591428083147352743781, −2.61091407349912333539671087887, −2.40364445678064681100619539383, −2.22939093840673703378916003036, −1.69125022609278132803550252304, −1.39696303171074248558626072289, −1.16398045912261658201966057244, 1.16398045912261658201966057244, 1.39696303171074248558626072289, 1.69125022609278132803550252304, 2.22939093840673703378916003036, 2.40364445678064681100619539383, 2.61091407349912333539671087887, 3.24311040591428083147352743781, 3.50891964819255802107469484730, 3.64982058917725737759659411604, 3.98320724455297996743241071986, 4.30193486698814366703086794389, 4.35923000731639677080293535208, 4.60589431097303281457343355029, 4.83894833729736448002146758436, 5.03023761593222638131806825818, 5.53338129185782295061768046643, 5.57059668384447078051761608503, 5.96629303561939938560409251139, 6.32769406293269554289061641139, 6.61642633441396966534873934476, 6.81732356364236332115922872824, 6.99549741217795448239305223620, 7.18470734165000147642600793946, 7.25497963733971226059976244055, 7.74965213848326785022799319923

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