Properties

Label 6-8550e3-1.1-c1e3-0-10
Degree $6$
Conductor $625026375000$
Sign $1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
Motivic weight $1$
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  + 3·2-s + 6·4-s + 2·7-s + 10·8-s + 5·11-s + 6·14-s + 15·16-s − 6·17-s + 3·19-s + 15·22-s + 23-s + 12·28-s + 15·29-s − 31-s + 21·32-s − 18·34-s − 4·37-s + 9·38-s + 6·41-s + 10·43-s + 30·44-s + 3·46-s − 7·49-s + 5·53-s + 20·56-s + 45·58-s + 12·59-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 0.755·7-s + 3.53·8-s + 1.50·11-s + 1.60·14-s + 15/4·16-s − 1.45·17-s + 0.688·19-s + 3.19·22-s + 0.208·23-s + 2.26·28-s + 2.78·29-s − 0.179·31-s + 3.71·32-s − 3.08·34-s − 0.657·37-s + 1.45·38-s + 0.937·41-s + 1.52·43-s + 4.52·44-s + 0.442·46-s − 49-s + 0.686·53-s + 2.67·56-s + 5.90·58-s + 1.56·59-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(49.65471187\)
\(L(\frac12)\) \(\approx\) \(49.65471187\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{3} \)
3 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 - 2 T + 11 T^{2} - 26 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 36 T^{2} - 107 T^{3} + 36 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T^{2} + 82 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 150 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 60 T^{2} - 49 T^{3} + 60 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 15 T + 144 T^{2} - 879 T^{3} + 144 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + T + 8 T^{2} - 47 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 95 T^{2} + 280 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 51 T^{2} - 168 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 157 T^{2} - 878 T^{3} + 157 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 117 T^{2} - 36 T^{3} + 117 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 102 T^{2} - 347 T^{3} + 102 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T - 27 T^{2} + 1050 T^{3} - 27 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 46 T^{2} + 517 T^{3} + 46 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - T + 154 T^{2} + 7 T^{3} + 154 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 24 T + 387 T^{2} - 3750 T^{3} + 387 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 9 T + 138 T^{2} - 773 T^{3} + 138 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 5 T + 224 T^{2} - 749 T^{3} + 224 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 11 T + 222 T^{2} - 1787 T^{3} + 222 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 23 T + 234 T^{2} - 1787 T^{3} + 234 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 14 T + 197 T^{2} - 1742 T^{3} + 197 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.71356411083531853867863444944, −6.54962312579035980686255608155, −6.35725201065930785945557507554, −6.23340956559213953648331700444, −6.00221017337040224056485351019, −5.66342727451410980038340385076, −5.32018264938890129114970788424, −5.11214989040029445347142915604, −4.95337014137384165271349781782, −4.77602086070496884053760557578, −4.52480889681891354924898888846, −4.30213120317229113320878845579, −4.08142468131472553532568346348, −3.67483620174666707327009780563, −3.62539071905029847227501644835, −3.44493295092591757962225673317, −2.95571960065060410089340804715, −2.75639612573205063574328014073, −2.42900083682186409095468063648, −2.10130147266389542894321434480, −1.88551383634954147353384526079, −1.82383135188140536533507806397, −0.880073733301537637760853003870, −0.858316979468417583863481948420, −0.837686227781796766820280731618, 0.837686227781796766820280731618, 0.858316979468417583863481948420, 0.880073733301537637760853003870, 1.82383135188140536533507806397, 1.88551383634954147353384526079, 2.10130147266389542894321434480, 2.42900083682186409095468063648, 2.75639612573205063574328014073, 2.95571960065060410089340804715, 3.44493295092591757962225673317, 3.62539071905029847227501644835, 3.67483620174666707327009780563, 4.08142468131472553532568346348, 4.30213120317229113320878845579, 4.52480889681891354924898888846, 4.77602086070496884053760557578, 4.95337014137384165271349781782, 5.11214989040029445347142915604, 5.32018264938890129114970788424, 5.66342727451410980038340385076, 6.00221017337040224056485351019, 6.23340956559213953648331700444, 6.35725201065930785945557507554, 6.54962312579035980686255608155, 6.71356411083531853867863444944

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