Properties

Label 6-2475e3-1.1-c1e3-0-3
Degree $6$
Conductor $15160921875$
Sign $1$
Analytic cond. $7718.92$
Root an. cond. $4.44555$
Motivic weight $1$
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  + 2·2-s + 3·4-s + 3·7-s + 4·8-s − 3·11-s + 5·13-s + 6·14-s + 3·16-s + 4·17-s − 19-s − 6·22-s + 10·26-s + 9·28-s − 2·29-s + 17·31-s + 6·32-s + 8·34-s − 2·38-s − 4·41-s + 17·43-s − 9·44-s + 10·47-s − 5·49-s + 15·52-s + 6·53-s + 12·56-s − 4·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.13·7-s + 1.41·8-s − 0.904·11-s + 1.38·13-s + 1.60·14-s + 3/4·16-s + 0.970·17-s − 0.229·19-s − 1.27·22-s + 1.96·26-s + 1.70·28-s − 0.371·29-s + 3.05·31-s + 1.06·32-s + 1.37·34-s − 0.324·38-s − 0.624·41-s + 2.59·43-s − 1.35·44-s + 1.45·47-s − 5/7·49-s + 2.08·52-s + 0.824·53-s + 1.60·56-s − 0.525·58-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(7718.92\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{6} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.78553830\)
\(L(\frac12)\) \(\approx\) \(15.78553830\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{3} \)
good2$D_{6}$ \( 1 - p T + T^{2} + p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 3 T + 2 p T^{2} - 25 T^{3} + 2 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 5 T + 3 p T^{2} - 122 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 26 T^{2} - 158 T^{3} + 26 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + T + 2 p T^{2} + 63 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 26 T^{2} + 58 T^{3} + 26 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 63 T^{2} + 132 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 36 T^{2} + 34 T^{3} + 36 p T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 4 T + 122 T^{2} + 326 T^{3} + 122 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 17 T + 97 T^{2} - 362 T^{3} + 97 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 10 T + 166 T^{2} - 948 T^{3} + 166 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 6 T + 11 T^{2} + 188 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 6 T + 146 T^{2} - 572 T^{3} + 146 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 3 T + 95 T^{2} + 10 p T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 7 T + 9 T^{2} + 650 T^{3} + 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 26 T + 430 T^{2} + 4272 T^{3} + 430 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 175 T^{2} + 988 T^{3} + 175 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T + 200 T^{2} - 736 T^{3} + 200 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 33 T^{2} - 4 p T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 2 T + 167 T^{2} + 28 T^{3} + 167 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 29 T + 366 T^{2} - 3473 T^{3} + 366 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.035083184425074321122808585108, −7.54937049818895415427816785601, −7.34674918441487602633326640143, −7.14088546016921195169468202611, −6.96025917029681205196735449243, −6.35738895911670387034565210788, −6.29477477082990396793266054897, −5.94031325246090108406684222580, −5.75033843238714488198867703511, −5.75004515543937225691441764321, −5.02469967909109632998065414571, −5.01094767125869889413754071595, −4.79706910050281371658088931911, −4.25074388172314136697070592345, −4.19835048271099079621864821606, −4.09197974434658888116327167296, −3.46811085292247000938630381903, −3.24983308252663908230571747431, −2.75466630462655591180226197173, −2.66355995427084745308497685213, −2.38985769805978353573635527817, −1.73138334807340817136522029027, −1.58848550648761436895596697014, −0.864882281999902894658317828280, −0.801515972272994065579618091862, 0.801515972272994065579618091862, 0.864882281999902894658317828280, 1.58848550648761436895596697014, 1.73138334807340817136522029027, 2.38985769805978353573635527817, 2.66355995427084745308497685213, 2.75466630462655591180226197173, 3.24983308252663908230571747431, 3.46811085292247000938630381903, 4.09197974434658888116327167296, 4.19835048271099079621864821606, 4.25074388172314136697070592345, 4.79706910050281371658088931911, 5.01094767125869889413754071595, 5.02469967909109632998065414571, 5.75004515543937225691441764321, 5.75033843238714488198867703511, 5.94031325246090108406684222580, 6.29477477082990396793266054897, 6.35738895911670387034565210788, 6.96025917029681205196735449243, 7.14088546016921195169468202611, 7.34674918441487602633326640143, 7.54937049818895415427816785601, 8.035083184425074321122808585108

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