Properties

Label 6-1805e3-1.1-c1e3-0-0
Degree $6$
Conductor $5880735125$
Sign $1$
Analytic cond. $2994.07$
Root an. cond. $3.79644$
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-s − 3-s + 4-s − 3·5-s + 6-s + 2·7-s + 2·8-s − 2·9-s + 3·10-s − 5·11-s − 12-s + 15·13-s − 2·14-s + 3·15-s − 5·16-s + 17-s + 2·18-s − 3·20-s − 2·21-s + 5·22-s + 4·23-s − 2·24-s + 6·25-s − 15·26-s + 6·27-s + 2·28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.755·7-s + 0.707·8-s − 2/3·9-s + 0.948·10-s − 1.50·11-s − 0.288·12-s + 4.16·13-s − 0.534·14-s + 0.774·15-s − 5/4·16-s + 0.242·17-s + 0.471·18-s − 0.670·20-s − 0.436·21-s + 1.06·22-s + 0.834·23-s − 0.408·24-s + 6/5·25-s − 2.94·26-s + 1.15·27-s + 0.377·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(5^{3} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2994.07\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 5^{3} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.448734346\)
\(L(\frac12)\) \(\approx\) \(1.448734346\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{3} \)
19 \( 1 \)
good2$A_4\times C_2$ \( 1 + T - 3 T^{3} + p^{2} T^{5} + p^{3} T^{6} \)
3$A_4\times C_2$ \( 1 + T + p T^{2} - T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 - 2 T + 16 T^{2} - 29 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 5 T + 35 T^{2} + 109 T^{3} + 35 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{3} \)
17$A_4\times C_2$ \( 1 - T + 7 T^{2} - 27 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 4 T + 30 T^{2} - 135 T^{3} + 30 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 2 T + 82 T^{2} - 117 T^{3} + 82 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - T + 87 T^{2} - 55 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 2 T - 8 T^{2} + 79 T^{3} - 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 2 T + 80 T^{2} - 127 T^{3} + 80 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + T + 85 T^{2} - 35 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 6 T + 134 T^{2} - 515 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 11 T + 117 T^{2} + 855 T^{3} + 117 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 6 T + 170 T^{2} + 659 T^{3} + 170 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 9 T + 134 T^{2} + 1049 T^{3} + 134 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 20 T + 309 T^{2} - 2768 T^{3} + 309 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 29 T + 449 T^{2} - 4585 T^{3} + 449 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 22 T + 336 T^{2} + 3289 T^{3} + 336 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 24 T + 353 T^{2} - 3544 T^{3} + 353 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 3 T + 195 T^{2} + 575 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 14 T + 231 T^{2} - 2436 T^{3} + 231 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 7 T + 225 T^{2} + 1237 T^{3} + 225 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.222989055078118401426871969071, −8.027045388757018179582003311430, −7.84218440493727340917441409738, −7.63302592154450424479559425516, −7.36947446937082273637432943860, −6.70548747859463901119183920022, −6.54825705798647467866056213418, −6.48424287429884955539389438704, −6.31751367391346621221876241436, −5.78318229225283275498511470231, −5.31007713684764805376829016097, −5.24524352116551532024253028374, −5.14809540494088647148545208749, −4.40821243399752888645252086632, −4.40019781541403609919521142526, −4.06919597428953661941068321254, −3.56331298910169455045925814089, −3.42506482079454655969003585468, −2.89244684749474001959868133557, −2.89225026118969653358759324495, −2.07894287110153800711277132316, −1.72543116089906707823967062571, −1.16585413015734357875497579948, −1.03007322032937494811053869284, −0.42493006675959888467559992385, 0.42493006675959888467559992385, 1.03007322032937494811053869284, 1.16585413015734357875497579948, 1.72543116089906707823967062571, 2.07894287110153800711277132316, 2.89225026118969653358759324495, 2.89244684749474001959868133557, 3.42506482079454655969003585468, 3.56331298910169455045925814089, 4.06919597428953661941068321254, 4.40019781541403609919521142526, 4.40821243399752888645252086632, 5.14809540494088647148545208749, 5.24524352116551532024253028374, 5.31007713684764805376829016097, 5.78318229225283275498511470231, 6.31751367391346621221876241436, 6.48424287429884955539389438704, 6.54825705798647467866056213418, 6.70548747859463901119183920022, 7.36947446937082273637432943860, 7.63302592154450424479559425516, 7.84218440493727340917441409738, 8.027045388757018179582003311430, 8.222989055078118401426871969071

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