Properties

Label 6-335e3-335.334-c0e3-0-1
Degree $6$
Conductor $37595375$
Sign $1$
Analytic cond. $0.00467310$
Root an. cond. $0.408884$
Motivic weight $0$
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  + 3·5-s − 8-s − 3·13-s − 3·19-s + 6·25-s − 27-s − 3·40-s − 9·65-s + 3·67-s − 3·71-s − 9·95-s + 3·104-s + 3·121-s + 10·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s + 3·152-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯
L(s)  = 1  + 3·5-s − 8-s − 3·13-s − 3·19-s + 6·25-s − 27-s − 3·40-s − 9·65-s + 3·67-s − 3·71-s − 9·95-s + 3·104-s + 3·121-s + 10·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s + 3·152-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37595375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37595375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(37595375\)    =    \(5^{3} \cdot 67^{3}\)
Sign: $1$
Analytic conductor: \(0.00467310\)
Root analytic conductor: \(0.408884\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{335} (334, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 37595375,\ (\ :0, 0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6413277561\)
\(L(\frac12)\) \(\approx\) \(0.6413277561\)
\(L(1)\) 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)$
bad5$C_1$ \( ( 1 - T )^{3} \)
67$C_1$ \( ( 1 - T )^{3} \)
good2$C_6$ \( 1 + T^{3} + T^{6} \)
3$C_6$ \( 1 + T^{3} + T^{6} \)
7$C_6$ \( 1 + T^{3} + T^{6} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
13$C_2$ \( ( 1 + T + T^{2} )^{3} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
19$C_2$ \( ( 1 + T + T^{2} )^{3} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
29$C_6$ \( 1 + T^{3} + T^{6} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
43$C_6$ \( 1 + T^{3} + T^{6} \)
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$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_2$ \( ( 1 + T + T^{2} )^{3} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 + T^{3} + T^{6} \)
97$C_6$ \( 1 + T^{3} + T^{6} \)
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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

−10.41798551668896672858128052855, −10.22902589876060478282363114307, −9.829114752824239301839579745165, −9.726120685350341513084077659429, −9.374343001607053144156478783777, −9.149549491802112850629565265460, −8.832056223902490822940548597385, −8.352929164953226007503362542103, −8.352446113432306906431761893840, −7.43804807196807448950154101411, −7.27354473602909558849204002413, −6.92443840811092655576461129943, −6.41843497820471896040389423874, −6.30570089708396643621497282690, −5.98012143876439665013731833059, −5.64131577602335512212995357024, −5.09596984702261536648052383836, −5.06977199406534976849675565564, −4.52874915155761929951101455354, −4.15281317513195459619515169215, −3.29338205648354503099323886350, −2.59638138134959920910827783067, −2.54225804609270401666256445757, −2.10045216621666588780373750316, −1.74773988046245978161081355269, 1.74773988046245978161081355269, 2.10045216621666588780373750316, 2.54225804609270401666256445757, 2.59638138134959920910827783067, 3.29338205648354503099323886350, 4.15281317513195459619515169215, 4.52874915155761929951101455354, 5.06977199406534976849675565564, 5.09596984702261536648052383836, 5.64131577602335512212995357024, 5.98012143876439665013731833059, 6.30570089708396643621497282690, 6.41843497820471896040389423874, 6.92443840811092655576461129943, 7.27354473602909558849204002413, 7.43804807196807448950154101411, 8.352446113432306906431761893840, 8.352929164953226007503362542103, 8.832056223902490822940548597385, 9.149549491802112850629565265460, 9.374343001607053144156478783777, 9.726120685350341513084077659429, 9.829114752824239301839579745165, 10.22902589876060478282363114307, 10.41798551668896672858128052855

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