Properties

Label 6-51e6-1.1-c1e3-0-3
Degree $6$
Conductor $17596287801$
Sign $-1$
Analytic cond. $8958.85$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 6·5-s + 8-s − 6·11-s + 6·13-s + 3·16-s + 18·20-s − 12·23-s + 12·25-s + 9·31-s − 6·32-s − 3·37-s − 6·40-s − 6·41-s + 15·43-s + 18·44-s + 21·47-s − 18·49-s − 18·52-s + 18·53-s + 36·55-s + 9·59-s − 3·61-s + 2·64-s − 36·65-s − 9·67-s − 21·71-s + ⋯
L(s)  = 1  − 3/2·4-s − 2.68·5-s + 0.353·8-s − 1.80·11-s + 1.66·13-s + 3/4·16-s + 4.02·20-s − 2.50·23-s + 12/5·25-s + 1.61·31-s − 1.06·32-s − 0.493·37-s − 0.948·40-s − 0.937·41-s + 2.28·43-s + 2.71·44-s + 3.06·47-s − 2.57·49-s − 2.49·52-s + 2.47·53-s + 4.85·55-s + 1.17·59-s − 0.384·61-s + 1/4·64-s − 4.46·65-s − 1.09·67-s − 2.49·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 17^{6}\)
Sign: $-1$
Analytic conductor: \(8958.85\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{6} \cdot 17^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 \)
good2$A_4\times C_2$ \( 1 + 3 T^{2} - T^{3} + 3 p T^{4} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 6 T + 24 T^{2} + 61 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 18 T^{2} - T^{3} + 18 p T^{4} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 6 T + 3 p T^{2} + 108 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$A_4\times C_2$ \( 1 - 6 T + 30 T^{2} - 85 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 54 T^{2} + T^{3} + 54 p T^{4} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 12 T + 108 T^{2} + 589 T^{3} + 108 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 78 T^{2} + 9 T^{3} + 78 p T^{4} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 9 T + 99 T^{2} - 505 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 3 T + 75 T^{2} + 95 T^{3} + 75 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 6 T + 96 T^{2} + 333 T^{3} + 96 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 15 T + 165 T^{2} - 1201 T^{3} + 165 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 21 T + 285 T^{2} - 2295 T^{3} + 285 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 18 T + 231 T^{2} - 1836 T^{3} + 231 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 9 T + 141 T^{2} - 891 T^{3} + 141 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 3 T + 177 T^{2} + 367 T^{3} + 177 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 9 T + 99 T^{2} + 917 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 21 T + 321 T^{2} + 2963 T^{3} + 321 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 21 T + 309 T^{2} + 39 p T^{3} + 309 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 3 T + 156 T^{2} - 687 T^{3} + 156 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 9 T + 192 T^{2} + 1565 T^{3} + 192 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 15 T + 225 T^{2} - 2057 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 6 T + 195 T^{2} - 740 T^{3} + 195 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.428560144667808981581117291291, −7.87941589292949073400254586133, −7.81497714191117384184659056787, −7.64664430997919530625681563960, −7.45353463500612669171059239445, −7.27587434929146877243057360750, −6.62007482222455422737362839200, −6.58953722315549995687934727943, −5.99682886829853509557880165714, −5.97103558250766146029034892550, −5.47415572103556894114790791208, −5.29022575801388633320208839956, −5.22363863607710777366308438598, −4.38962141397053915080174541825, −4.35421620549798529336860977905, −4.35165262970577326395569248185, −3.87556710723459603829448189530, −3.78406056128829644051309521379, −3.77042010906190933378635847676, −2.98922568105921940236235900647, −2.75011465673255375323945064576, −2.53224915614440313639885323711, −1.93814225290713998147166399351, −1.23116187311688980868436746001, −1.04735991094932470943355931942, 0, 0, 0, 1.04735991094932470943355931942, 1.23116187311688980868436746001, 1.93814225290713998147166399351, 2.53224915614440313639885323711, 2.75011465673255375323945064576, 2.98922568105921940236235900647, 3.77042010906190933378635847676, 3.78406056128829644051309521379, 3.87556710723459603829448189530, 4.35165262970577326395569248185, 4.35421620549798529336860977905, 4.38962141397053915080174541825, 5.22363863607710777366308438598, 5.29022575801388633320208839956, 5.47415572103556894114790791208, 5.97103558250766146029034892550, 5.99682886829853509557880165714, 6.58953722315549995687934727943, 6.62007482222455422737362839200, 7.27587434929146877243057360750, 7.45353463500612669171059239445, 7.64664430997919530625681563960, 7.81497714191117384184659056787, 7.87941589292949073400254586133, 8.428560144667808981581117291291

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