Properties

Label 6-3380e3-1.1-c1e3-0-2
Degree $6$
Conductor $38614472000$
Sign $-1$
Analytic cond. $19659.9$
Root an. cond. $5.19513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 3·9-s − 6·11-s + 6·23-s + 6·25-s + 2·27-s − 6·29-s − 6·31-s − 12·37-s + 6·41-s − 6·43-s + 9·45-s − 12·47-s − 9·49-s − 6·53-s + 18·55-s + 24·59-s − 6·61-s + 12·67-s − 24·73-s − 12·79-s − 12·83-s − 24·89-s − 18·97-s + 18·99-s − 6·101-s − 12·103-s + ⋯
L(s)  = 1  − 1.34·5-s − 9-s − 1.80·11-s + 1.25·23-s + 6/5·25-s + 0.384·27-s − 1.11·29-s − 1.07·31-s − 1.97·37-s + 0.937·41-s − 0.914·43-s + 1.34·45-s − 1.75·47-s − 9/7·49-s − 0.824·53-s + 2.42·55-s + 3.12·59-s − 0.768·61-s + 1.46·67-s − 2.80·73-s − 1.35·79-s − 1.31·83-s − 2.54·89-s − 1.82·97-s + 1.80·99-s − 0.597·101-s − 1.18·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 5^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(19659.9\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3380} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{6} \cdot 5^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good3$S_4\times C_2$ \( 1 + p T^{2} - 2 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 9 T^{2} - 12 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T + 3 p T^{2} + 114 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 9 T^{2} - 6 p T^{3} + 9 p T^{4} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 6 T + 39 T^{2} - 102 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 6 T + 75 T^{2} + 264 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 6 T + 81 T^{2} + 306 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 636 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 87 T^{2} - 420 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 87 T^{2} + 470 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 12 T + 105 T^{2} + 660 T^{3} + 105 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 75 T^{2} + 660 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 24 T + 357 T^{2} - 3246 T^{3} + 357 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 87 T^{2} + 200 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 12 T + 153 T^{2} - 1020 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 3 T^{2} + 702 T^{3} - 3 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 24 T + 363 T^{2} + 3756 T^{3} + 363 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 12 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 1740 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 24 T + 3 p T^{2} + 2256 T^{3} + 3 p^{2} T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 18 T + 351 T^{2} + 3516 T^{3} + 351 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.146536950362588763111214233918, −7.74695514290728083945438285874, −7.58964389050066241456294460628, −7.23623588741606551383490496820, −6.89604110434204884832650618622, −6.86528319203854846151104059757, −6.84145696827528899800832713178, −6.22670341573365796696457829565, −5.76907972313469933364996967159, −5.70629146729204416269836977275, −5.31508567230077005618969981159, −5.30773643841967203561073081225, −4.98258150634891451428981665088, −4.71545020971716465221253076056, −4.31255494357185134429211568235, −4.12512284293822952311782483290, −3.69538610509362182223665025484, −3.42775434502845963608355215715, −3.21405660056192398071228446178, −2.86455249044340916088909441166, −2.70539153527588991816632057526, −2.39953244332321999250912184705, −1.75731991098996895351348107208, −1.51780181829343858783964529183, −1.04606007879802810539920122569, 0, 0, 0, 1.04606007879802810539920122569, 1.51780181829343858783964529183, 1.75731991098996895351348107208, 2.39953244332321999250912184705, 2.70539153527588991816632057526, 2.86455249044340916088909441166, 3.21405660056192398071228446178, 3.42775434502845963608355215715, 3.69538610509362182223665025484, 4.12512284293822952311782483290, 4.31255494357185134429211568235, 4.71545020971716465221253076056, 4.98258150634891451428981665088, 5.30773643841967203561073081225, 5.31508567230077005618969981159, 5.70629146729204416269836977275, 5.76907972313469933364996967159, 6.22670341573365796696457829565, 6.84145696827528899800832713178, 6.86528319203854846151104059757, 6.89604110434204884832650618622, 7.23623588741606551383490496820, 7.58964389050066241456294460628, 7.74695514290728083945438285874, 8.146536950362588763111214233918

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