Properties

Label 6-135e3-1.1-c3e3-0-2
Degree $6$
Conductor $2460375$
Sign $1$
Analytic cond. $505.358$
Root an. cond. $2.82227$
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-s + 15·5-s + 44·7-s − 9·8-s − 15·10-s + 38·11-s + 28·13-s − 44·14-s + 11·16-s − 19·17-s + 187·19-s − 38·22-s − 81·23-s + 150·25-s − 28·26-s + 160·29-s + 227·31-s − 58·32-s + 19·34-s + 660·35-s + 78·37-s − 187·38-s − 135·40-s − 338·41-s + 22·43-s + 81·46-s − 472·47-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.34·5-s + 2.37·7-s − 0.397·8-s − 0.474·10-s + 1.04·11-s + 0.597·13-s − 0.839·14-s + 0.171·16-s − 0.271·17-s + 2.25·19-s − 0.368·22-s − 0.734·23-s + 6/5·25-s − 0.211·26-s + 1.02·29-s + 1.31·31-s − 0.320·32-s + 0.0958·34-s + 3.18·35-s + 0.346·37-s − 0.798·38-s − 0.533·40-s − 1.28·41-s + 0.0780·43-s + 0.259·46-s − 1.46·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2460375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2460375 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2460375\)    =    \(3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(505.358\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{135} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2460375,\ (\ :3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.502362160\)
\(L(\frac12)\) \(\approx\) \(5.502362160\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + T^{2} + 5 p T^{3} + p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 - 44 T + 1581 T^{2} - 31984 T^{3} + 1581 p^{3} T^{4} - 44 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 38 T + 1381 T^{2} - 17876 T^{3} + 1381 p^{3} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 28 T + 155 p T^{2} - 22912 T^{3} + 155 p^{4} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 19 T + 3262 T^{2} - 367193 T^{3} + 3262 p^{3} T^{4} + 19 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 187 T + 24164 T^{2} - 2039395 T^{3} + 24164 p^{3} T^{4} - 187 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 81 T + 13200 T^{2} - 72927 T^{3} + 13200 p^{3} T^{4} + 81 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 160 T + 25399 T^{2} + 88280 T^{3} + 25399 p^{3} T^{4} - 160 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 227 T + 71400 T^{2} - 13771435 T^{3} + 71400 p^{3} T^{4} - 227 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 78 T + 52035 T^{2} + 5735212 T^{3} + 52035 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 338 T + 163951 T^{2} + 34473956 T^{3} + 163951 p^{3} T^{4} + 338 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 22 T + 78605 T^{2} + 14966252 T^{3} + 78605 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 472 T + 365677 T^{2} + 97725712 T^{3} + 365677 p^{3} T^{4} + 472 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 521 T + 508018 T^{2} - 154190045 T^{3} + 508018 p^{3} T^{4} - 521 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 140 T + 449689 T^{2} - 23374640 T^{3} + 449689 p^{3} T^{4} - 140 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 595 T + 678194 T^{2} - 268324783 T^{3} + 678194 p^{3} T^{4} - 595 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 878 T + 978237 T^{2} - 516844828 T^{3} + 978237 p^{3} T^{4} - 878 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 602 T + 490081 T^{2} + 150373964 T^{3} + 490081 p^{3} T^{4} + 602 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 1294 T + 1071143 T^{2} - 602684716 T^{3} + 1071143 p^{3} T^{4} - 1294 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 629 T + 1576176 T^{2} - 622253365 T^{3} + 1576176 p^{3} T^{4} - 629 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 1287 T + 1349484 T^{2} + 1125375327 T^{3} + 1349484 p^{3} T^{4} + 1287 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 2154 T + 3172479 T^{2} - 3111332052 T^{3} + 3172479 p^{3} T^{4} - 2154 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 1392 T + 3281955 T^{2} - 2604477152 T^{3} + 3281955 p^{3} T^{4} - 1392 p^{6} T^{5} + p^{9} T^{6} \)
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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

−11.38871870626451435951230400215, −11.28788386475852586861972154692, −10.52834706060518849615971317137, −10.47857349900496496955351377503, −9.748782798612067164587419733807, −9.740625732786603178422860955747, −9.440195046436596098741513155775, −8.748874984565395795904699133598, −8.498341799432472973639287371518, −8.387113389414034101231407143757, −7.79568480494185132574097298263, −7.58604885732771397060705404899, −6.77982788961548332106954320230, −6.68119614195594893227291569658, −6.07658808729188560518615531329, −5.78115447342691288264749105877, −5.03793412252658427284035642149, −4.99528109092923355629027008038, −4.64732004912900678679289690576, −3.69474268241816453911910736652, −3.37523131570203269400819655465, −2.41680980255820951534523683057, −1.96515825962907054043415800713, −1.16991633633344706421380871163, −1.11031820612410364928005082638, 1.11031820612410364928005082638, 1.16991633633344706421380871163, 1.96515825962907054043415800713, 2.41680980255820951534523683057, 3.37523131570203269400819655465, 3.69474268241816453911910736652, 4.64732004912900678679289690576, 4.99528109092923355629027008038, 5.03793412252658427284035642149, 5.78115447342691288264749105877, 6.07658808729188560518615531329, 6.68119614195594893227291569658, 6.77982788961548332106954320230, 7.58604885732771397060705404899, 7.79568480494185132574097298263, 8.387113389414034101231407143757, 8.498341799432472973639287371518, 8.748874984565395795904699133598, 9.440195046436596098741513155775, 9.740625732786603178422860955747, 9.748782798612067164587419733807, 10.47857349900496496955351377503, 10.52834706060518849615971317137, 11.28788386475852586861972154692, 11.38871870626451435951230400215

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