Properties

Label 6-4704e3-1.1-c1e3-0-1
Degree $6$
Conductor $104088305664$
Sign $1$
Analytic cond. $52994.8$
Root an. cond. $6.12875$
Motivic weight $1$
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  − 3·3-s + 6·9-s + 3·13-s − 6·17-s − 3·19-s − 6·23-s − 6·25-s − 10·27-s + 12·29-s − 3·31-s + 3·37-s − 9·39-s + 6·41-s + 15·43-s − 12·47-s + 18·51-s + 6·53-s + 9·57-s − 12·59-s + 18·61-s + 9·67-s + 18·69-s − 33·73-s + 18·75-s − 27·79-s + 15·81-s − 18·83-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.832·13-s − 1.45·17-s − 0.688·19-s − 1.25·23-s − 6/5·25-s − 1.92·27-s + 2.22·29-s − 0.538·31-s + 0.493·37-s − 1.44·39-s + 0.937·41-s + 2.28·43-s − 1.75·47-s + 2.52·51-s + 0.824·53-s + 1.19·57-s − 1.56·59-s + 2.30·61-s + 1.09·67-s + 2.16·69-s − 3.86·73-s + 2.07·75-s − 3.03·79-s + 5/3·81-s − 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{15} \cdot 3^{3} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(52994.8\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4704} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{15} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.348340772\)
\(L(\frac12)\) \(\approx\) \(1.348340772\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
7 \( 1 \)
good5$S_4\times C_2$ \( 1 + 6 T^{2} - 4 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T^{2} + 38 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 6 T + 27 T^{2} + 108 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 21 T^{2} + 2 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 12 T + 126 T^{2} - 728 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 3 T + 72 T^{2} + 139 T^{3} + 72 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 3 T + 27 T^{2} + 146 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 27 T^{2} + 20 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 15 T + 177 T^{2} - 1318 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 12 T + 153 T^{2} + 1016 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 6 T + 78 T^{2} - 802 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T + 198 T^{2} + 1334 T^{3} + 198 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 18 T + 195 T^{2} - 1644 T^{3} + 195 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 9 T + 189 T^{2} - 1042 T^{3} + 189 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 177 T^{2} - 32 T^{3} + 177 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 33 T + 555 T^{2} + 5814 T^{3} + 555 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 27 T + 432 T^{2} + 4619 T^{3} + 432 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 18 T + 234 T^{2} + 2040 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 12 T + 279 T^{2} - 2024 T^{3} + 279 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 264 T^{2} - 38 T^{3} + 264 p T^{4} + p^{3} T^{6} \)
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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

−7.34147004518892955097842480749, −6.99163443732673440325452421746, −6.69376291845125105244375062946, −6.64637730683348267657902414434, −6.18679150991610316101451484892, −6.07709644814480298072866209082, −6.05820600009165141562909258339, −5.60748839762312218104233348494, −5.51274794169166703435755053520, −5.26073791843741128494996673617, −4.70157213118691035747950640750, −4.50062880667888497807274704531, −4.44636124073174718093268599511, −4.05426619709161814947418281730, −3.98456785623740807446332889337, −3.82214350692409578491386074162, −2.93057377812247169875192048476, −2.82883155546722697079469801715, −2.81316858823045985976162914931, −1.93626380389977736176338894341, −1.89696302160798495023872349741, −1.64419091416407840581965803806, −1.06390329268472337533951301534, −0.55544591403100037996035941053, −0.38536781892393327736767466535, 0.38536781892393327736767466535, 0.55544591403100037996035941053, 1.06390329268472337533951301534, 1.64419091416407840581965803806, 1.89696302160798495023872349741, 1.93626380389977736176338894341, 2.81316858823045985976162914931, 2.82883155546722697079469801715, 2.93057377812247169875192048476, 3.82214350692409578491386074162, 3.98456785623740807446332889337, 4.05426619709161814947418281730, 4.44636124073174718093268599511, 4.50062880667888497807274704531, 4.70157213118691035747950640750, 5.26073791843741128494996673617, 5.51274794169166703435755053520, 5.60748839762312218104233348494, 6.05820600009165141562909258339, 6.07709644814480298072866209082, 6.18679150991610316101451484892, 6.64637730683348267657902414434, 6.69376291845125105244375062946, 6.99163443732673440325452421746, 7.34147004518892955097842480749

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