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Properties

Label 6-1607e3-1607.1606-c0e3-0-0
Degree $6$
Conductor $4149995543$
Sign $1$
Analytic cond. $0.515844$
Root an. cond. $0.895543$
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 + 3·4-s + 2·8-s − 9·16-s + 3·25-s − 27-s + 9·32-s + 3·49-s − 9·50-s + 3·54-s + 3·64-s + 6·73-s − 9·98-s + 9·100-s − 3·101-s − 3·108-s − 3·113-s + 3·121-s + 127-s − 18·128-s + 131-s + 137-s + 139-s − 18·146-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 3·2-s + 3·4-s + 2·8-s − 9·16-s + 3·25-s − 27-s + 9·32-s + 3·49-s − 9·50-s + 3·54-s + 3·64-s + 6·73-s − 9·98-s + 9·100-s − 3·101-s − 3·108-s − 3·113-s + 3·121-s + 127-s − 18·128-s + 131-s + 137-s + 139-s − 18·146-s + 149-s + 151-s + 157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1607^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1607^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(1607^{3}\)
Sign: $1$
Analytic conductor: \(0.515844\)
Root analytic conductor: \(0.895543\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1607} (1606, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 1607^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1879845889\)
\(L(\frac12)\) \(\approx\) \(0.1879845889\)
\(L(1)\) 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)$
bad1607$C_1$ \( ( 1 - T )^{3} \)
good2$C_2$ \( ( 1 + T + T^{2} )^{3} \)
3$C_6$ \( 1 + T^{3} + T^{6} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
17$C_6$ \( 1 + T^{3} + T^{6} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
23$C_6$ \( 1 + T^{3} + T^{6} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
31$C_6$ \( 1 + T^{3} + T^{6} \)
37$C_6$ \( 1 + T^{3} + T^{6} \)
41$C_6$ \( 1 + T^{3} + T^{6} \)
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$C_6$ \( 1 + T^{3} + T^{6} \)
59$C_6$ \( 1 + T^{3} + T^{6} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
67$C_6$ \( 1 + T^{3} + T^{6} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_1$ \( ( 1 - T )^{6} \)
79$C_6$ \( 1 + T^{3} + T^{6} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_6$ \( 1 + T^{3} + T^{6} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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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

−8.742640672693309048328519792243, −8.298227318307001574296216918487, −8.135943986621417611381265356426, −7.964826619218766184799269713164, −7.71478181047766987806309191698, −7.43854297173943735193113827235, −7.02975673050162125746815794263, −6.81941381538044993647227703278, −6.65393576430579677558700052760, −6.58586074109333343961397607992, −5.51122618681680652642453779464, −5.49080946603673185456856038042, −5.45924739543608903794491235953, −4.76660485996989065534298728072, −4.67769412317482035968108946283, −4.31194667222658466989612763077, −4.06553163790950539769540800055, −3.49458467775336839900947188975, −3.45562051692244804622482335829, −2.45170894034929625963432419758, −2.43669061160438069120830602577, −1.96913379618923402435321359855, −1.33268396937832234704938316805, −1.00162124443444014944302860188, −0.68365409226508332719049000909, 0.68365409226508332719049000909, 1.00162124443444014944302860188, 1.33268396937832234704938316805, 1.96913379618923402435321359855, 2.43669061160438069120830602577, 2.45170894034929625963432419758, 3.45562051692244804622482335829, 3.49458467775336839900947188975, 4.06553163790950539769540800055, 4.31194667222658466989612763077, 4.67769412317482035968108946283, 4.76660485996989065534298728072, 5.45924739543608903794491235953, 5.49080946603673185456856038042, 5.51122618681680652642453779464, 6.58586074109333343961397607992, 6.65393576430579677558700052760, 6.81941381538044993647227703278, 7.02975673050162125746815794263, 7.43854297173943735193113827235, 7.71478181047766987806309191698, 7.964826619218766184799269713164, 8.135943986621417611381265356426, 8.298227318307001574296216918487, 8.742640672693309048328519792243

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