Properties

Label 6-370e3-1.1-c1e3-0-0
Degree $6$
Conductor $50653000$
Sign $1$
Analytic cond. $25.7891$
Root an. cond. $1.71885$
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  + 3·2-s + 6·4-s − 3·5-s − 7-s + 10·8-s + 9-s − 9·10-s + 11·11-s − 3·14-s + 15·16-s + 17-s + 3·18-s − 18·20-s + 33·22-s − 2·23-s + 6·25-s − 4·27-s − 6·28-s − 5·29-s + 3·31-s + 21·32-s + 3·34-s + 3·35-s + 6·36-s − 3·37-s − 30·40-s − 9·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.34·5-s − 0.377·7-s + 3.53·8-s + 1/3·9-s − 2.84·10-s + 3.31·11-s − 0.801·14-s + 15/4·16-s + 0.242·17-s + 0.707·18-s − 4.02·20-s + 7.03·22-s − 0.417·23-s + 6/5·25-s − 0.769·27-s − 1.13·28-s − 0.928·29-s + 0.538·31-s + 3.71·32-s + 0.514·34-s + 0.507·35-s + 36-s − 0.493·37-s − 4.74·40-s − 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50653000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50653000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(50653000\)    =    \(2^{3} \cdot 5^{3} \cdot 37^{3}\)
Sign: $1$
Analytic conductor: \(25.7891\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 50653000,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.126132790\)
\(L(\frac12)\) \(\approx\) \(7.126132790\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
37$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 - T^{2} + 4 T^{3} - p T^{4} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + T + 13 T^{2} + 4 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - p T + 61 T^{2} - 234 T^{3} + 61 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - T^{2} - 32 T^{3} - p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - T + 39 T^{2} - 42 T^{3} + 39 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 47 T^{2} + 4 T^{3} + 47 p T^{4} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 2 T + 21 T^{2} + 156 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 5 T + 71 T^{2} + 214 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 21 T^{2} + 84 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 9 T + 83 T^{2} + 374 T^{3} + 83 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 11 T + 145 T^{2} + 866 T^{3} + 145 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 111 T^{2} - 132 T^{3} + 111 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 21 T + 239 T^{2} - 1910 T^{3} + 239 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 14 T + 211 T^{2} - 1572 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 7 T + 175 T^{2} - 850 T^{3} + 175 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 2 T + 171 T^{2} - 212 T^{3} + 171 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 22 T + 277 T^{2} - 2484 T^{3} + 277 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 16 T + 255 T^{2} + 2128 T^{3} + 255 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 26 T + 407 T^{2} + 4332 T^{3} + 407 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 2 T + 91 T^{2} - 12 p T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
97$S_4\times C_2$ \( 1 + 21 T + 371 T^{2} + 3758 T^{3} + 371 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.01293868207377400237488112873, −9.996826795763793106699149854426, −9.815362902437729114919347854157, −9.326477109673722951851686346505, −8.649053720970899898583072647572, −8.637363163196287707705750464790, −8.404430912473195544754393272519, −7.67374705854895532338206160241, −7.54712931157180239907265987070, −6.93467493168068283699568623376, −6.81645835521015055774884096073, −6.59182408600691790487324255180, −6.51753248776819812170403466852, −5.76275040668198468447716813559, −5.40238730863292871205504617186, −5.32531411687525139171609671547, −4.41345416412251328155331739465, −4.34861644169861943769931646905, −4.02831704671153362193075685414, −3.70727608921289895429051350715, −3.36064582233434907225880944563, −3.20876494903941789673814290933, −2.19114773505198629580624255052, −1.73928609384203473820565154826, −1.08962700757098009902654475464, 1.08962700757098009902654475464, 1.73928609384203473820565154826, 2.19114773505198629580624255052, 3.20876494903941789673814290933, 3.36064582233434907225880944563, 3.70727608921289895429051350715, 4.02831704671153362193075685414, 4.34861644169861943769931646905, 4.41345416412251328155331739465, 5.32531411687525139171609671547, 5.40238730863292871205504617186, 5.76275040668198468447716813559, 6.51753248776819812170403466852, 6.59182408600691790487324255180, 6.81645835521015055774884096073, 6.93467493168068283699568623376, 7.54712931157180239907265987070, 7.67374705854895532338206160241, 8.404430912473195544754393272519, 8.637363163196287707705750464790, 8.649053720970899898583072647572, 9.326477109673722951851686346505, 9.815362902437729114919347854157, 9.996826795763793106699149854426, 10.01293868207377400237488112873

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