Properties

Label 6-825e3-1.1-c1e3-0-3
Degree $6$
Conductor $561515625$
Sign $1$
Analytic cond. $285.886$
Root an. cond. $2.56664$
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 − 3·3-s + 4·4-s − 9·6-s + 8·7-s + 4·8-s + 6·9-s − 3·11-s − 12·12-s + 6·13-s + 24·14-s + 3·16-s + 4·17-s + 18·18-s − 2·19-s − 24·21-s − 9·22-s + 12·23-s − 12·24-s + 18·26-s − 10·27-s + 32·28-s − 8·29-s + 8·31-s − 32-s + 9·33-s + 12·34-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 2·4-s − 3.67·6-s + 3.02·7-s + 1.41·8-s + 2·9-s − 0.904·11-s − 3.46·12-s + 1.66·13-s + 6.41·14-s + 3/4·16-s + 0.970·17-s + 4.24·18-s − 0.458·19-s − 5.23·21-s − 1.91·22-s + 2.50·23-s − 2.44·24-s + 3.53·26-s − 1.92·27-s + 6.04·28-s − 1.48·29-s + 1.43·31-s − 0.176·32-s + 1.56·33-s + 2.05·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(285.886\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.544148764\)
\(L(\frac12)\) \(\approx\) \(8.544148764\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 23 T^{2} - 20 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 432 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 624 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 8 T + 145 T^{2} - 692 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 125 T^{2} - 592 T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 8 T + 127 T^{2} + 576 T^{3} + 127 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 12 T + 185 T^{2} - 1288 T^{3} + 185 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 18 T + 279 T^{2} - 2536 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 2 T + 245 T^{2} + 328 T^{3} + 245 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 255 T^{2} - 348 T^{3} + 255 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 8 T + 259 T^{2} + 1424 T^{3} + 259 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.168836514074333158517830630843, −8.535774714872927885589435672188, −8.511226800400000433009649511041, −8.042429170907351404348630653450, −7.978377094133140898193965544623, −7.47731373788340706321215657623, −7.34550030273298068337234858901, −6.96569658306558056074072397097, −6.47542481197842590998376621368, −6.38739929905039578222873250661, −5.81889204724798485412087037558, −5.58032556971060338129616761729, −5.39471893138383905450016926700, −5.07815310897846657421056372475, −4.98651758517345396645537036177, −4.67666615259883987453194053627, −4.28930424590553349006082575792, −4.16631700215969944884114739038, −3.79848399739394893566659218010, −3.16781990644841169311502821444, −2.88147475175510917268599223343, −2.08486401237106693669829180021, −1.75765100409066135275359600174, −1.16785437323250681236653867749, −0.963447407139270182895892958408, 0.963447407139270182895892958408, 1.16785437323250681236653867749, 1.75765100409066135275359600174, 2.08486401237106693669829180021, 2.88147475175510917268599223343, 3.16781990644841169311502821444, 3.79848399739394893566659218010, 4.16631700215969944884114739038, 4.28930424590553349006082575792, 4.67666615259883987453194053627, 4.98651758517345396645537036177, 5.07815310897846657421056372475, 5.39471893138383905450016926700, 5.58032556971060338129616761729, 5.81889204724798485412087037558, 6.38739929905039578222873250661, 6.47542481197842590998376621368, 6.96569658306558056074072397097, 7.34550030273298068337234858901, 7.47731373788340706321215657623, 7.978377094133140898193965544623, 8.042429170907351404348630653450, 8.511226800400000433009649511041, 8.535774714872927885589435672188, 9.168836514074333158517830630843

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