Properties

Label 6-1815e3-1.1-c3e3-0-1
Degree $6$
Conductor $5979018375$
Sign $1$
Analytic cond. $1.22808\times 10^{6}$
Root an. cond. $10.3483$
Motivic weight $3$
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  + 2·2-s + 9·3-s + 5·4-s − 15·5-s + 18·6-s − 10·7-s − 4·8-s + 54·9-s − 30·10-s + 45·12-s − 114·13-s − 20·14-s − 135·15-s − 27·16-s + 104·17-s + 108·18-s + 58·19-s − 75·20-s − 90·21-s + 120·23-s − 36·24-s + 150·25-s − 228·26-s + 270·27-s − 50·28-s + 220·29-s − 270·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 5/8·4-s − 1.34·5-s + 1.22·6-s − 0.539·7-s − 0.176·8-s + 2·9-s − 0.948·10-s + 1.08·12-s − 2.43·13-s − 0.381·14-s − 2.32·15-s − 0.421·16-s + 1.48·17-s + 1.41·18-s + 0.700·19-s − 0.838·20-s − 0.935·21-s + 1.08·23-s − 0.306·24-s + 6/5·25-s − 1.71·26-s + 1.92·27-s − 0.337·28-s + 1.40·29-s − 1.64·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(1.22808\times 10^{6}\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(16.75731072\)
\(L(\frac12)\) \(\approx\) \(16.75731072\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{3} \)
5$C_1$ \( ( 1 + p T )^{3} \)
11 \( 1 \)
good2$S_4\times C_2$ \( 1 - p T - T^{2} + p^{4} T^{3} - p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 10 T + 425 T^{2} + 3412 T^{3} + 425 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 114 T + 10303 T^{2} + 538132 T^{3} + 10303 p^{3} T^{4} + 114 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 104 T + 17531 T^{2} - 1030352 T^{3} + 17531 p^{3} T^{4} - 104 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 58 T + 9081 T^{2} - 861164 T^{3} + 9081 p^{3} T^{4} - 58 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 120 T + 25765 T^{2} - 2771856 T^{3} + 25765 p^{3} T^{4} - 120 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 220 T + 58859 T^{2} - 10101400 T^{3} + 58859 p^{3} T^{4} - 220 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 p T + 50781 T^{2} - 5187088 T^{3} + 50781 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 838 T + 375507 T^{2} - 103501764 T^{3} + 375507 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 156 T + 100167 T^{2} + 24516984 T^{3} + 100167 p^{3} T^{4} + 156 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 122 T + 193581 T^{2} + 17954308 T^{3} + 193581 p^{3} T^{4} + 122 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 504 T + 393661 T^{2} - 109025808 T^{3} + 393661 p^{3} T^{4} - 504 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 282 T + 224563 T^{2} - 87620892 T^{3} + 224563 p^{3} T^{4} - 282 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 548 T + 11419 p T^{2} - 226302104 T^{3} + 11419 p^{4} T^{4} - 548 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 414 T - 389 T^{2} - 154404524 T^{3} - 389 p^{3} T^{4} + 414 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 428 T + 695537 T^{2} + 249317576 T^{3} + 695537 p^{3} T^{4} + 428 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 912 T + 1171621 T^{2} + 649961952 T^{3} + 1171621 p^{3} T^{4} + 912 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 618 T + 1066507 T^{2} + 454366420 T^{3} + 1066507 p^{3} T^{4} + 618 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 542 T + 1317893 T^{2} - 445950836 T^{3} + 1317893 p^{3} T^{4} - 542 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 624021 T^{2} + 434328048 T^{3} + 624021 p^{3} T^{4} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 790 T - 3625 T^{2} + 827778380 T^{3} - 3625 p^{3} T^{4} - 790 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 2074 T + 3705231 T^{2} - 3687692268 T^{3} + 3705231 p^{3} T^{4} - 2074 p^{6} T^{5} + p^{9} T^{6} \)
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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

−7.68155212738953520809810581499, −7.56538731061693665212905206945, −7.50155607756184202835593213827, −7.26307822358259254531316984862, −6.93837220696184295005558493422, −6.48706647654754191234602591050, −6.33738140739165496773939797422, −6.13353153738808215739894937211, −5.58428701515979438719512434920, −5.33969528692438469881444665079, −4.78658643954611606608651611457, −4.68295404203727403511285468873, −4.58896284724051986410151801347, −4.22751270704659360203077457517, −3.81745474533621915538725886623, −3.50939886386724847681186909810, −3.00685929005974086106707187204, −2.99298786691303386945408748121, −2.75192513907620275509136818853, −2.64072958350920720113026199117, −2.15098337963056149801790933048, −1.61778701123934422766480284896, −0.977004267892786146990232190982, −0.68932957538676301339132156075, −0.53196175745247068709289621720, 0.53196175745247068709289621720, 0.68932957538676301339132156075, 0.977004267892786146990232190982, 1.61778701123934422766480284896, 2.15098337963056149801790933048, 2.64072958350920720113026199117, 2.75192513907620275509136818853, 2.99298786691303386945408748121, 3.00685929005974086106707187204, 3.50939886386724847681186909810, 3.81745474533621915538725886623, 4.22751270704659360203077457517, 4.58896284724051986410151801347, 4.68295404203727403511285468873, 4.78658643954611606608651611457, 5.33969528692438469881444665079, 5.58428701515979438719512434920, 6.13353153738808215739894937211, 6.33738140739165496773939797422, 6.48706647654754191234602591050, 6.93837220696184295005558493422, 7.26307822358259254531316984862, 7.50155607756184202835593213827, 7.56538731061693665212905206945, 7.68155212738953520809810581499

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