Properties

Label 6-8550e3-1.1-c1e3-0-16
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 $3$

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 − 4·13-s − 6·14-s + 15·16-s + 2·17-s − 3·19-s − 15·22-s − 23-s − 12·26-s − 12·28-s − 5·29-s + 5·31-s + 21·32-s + 6·34-s − 4·37-s − 9·38-s − 10·41-s − 2·43-s − 30·44-s − 3·46-s + 4·47-s − 11·49-s − 24·52-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 0.755·7-s + 3.53·8-s − 1.50·11-s − 1.10·13-s − 1.60·14-s + 15/4·16-s + 0.485·17-s − 0.688·19-s − 3.19·22-s − 0.208·23-s − 2.35·26-s − 2.26·28-s − 0.928·29-s + 0.898·31-s + 3.71·32-s + 1.02·34-s − 0.657·37-s − 1.45·38-s − 1.56·41-s − 0.304·43-s − 4.52·44-s − 0.442·46-s + 0.583·47-s − 1.57·49-s − 3.32·52-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: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{3} \)
3 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 + 2 T + 15 T^{2} + 18 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 5 T + 28 T^{2} + 83 T^{3} + 28 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 31 T^{2} + 98 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 45 T^{2} - 58 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + T + 24 T^{2} - 35 T^{3} + 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 5 T + 44 T^{2} + 93 T^{3} + 44 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 72 T^{2} - 309 T^{3} + 72 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 63 T^{2} + 152 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 147 T^{2} + 824 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T + 29 T^{2} + 334 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 4 T + 53 T^{2} - 580 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 86 T^{2} - 647 T^{3} + 86 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T + 205 T^{2} + 1366 T^{3} + 205 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 102 T^{2} + 153 T^{3} + 102 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 58 T^{2} + 361 T^{3} + 58 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 16 T + 215 T^{2} + 1918 T^{3} + 215 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 15 T + 210 T^{2} + 1947 T^{3} + 210 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T + 136 T^{2} + 713 T^{3} + 136 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 3 T + 214 T^{2} + 371 T^{3} + 214 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 9 T + 210 T^{2} + 1325 T^{3} + 210 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 237 T^{2} - 894 T^{3} + 237 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.08934295705747738318694380838, −6.80446632040736850083736512314, −6.68863249250605218438765759024, −6.64437403536975662372635464262, −6.04819064830567895615015633523, −5.93510110653598198001483741427, −5.90693115706123916205285929007, −5.50420412663478528617957155201, −5.25197185070482291066827672635, −5.19512525411581451122285830797, −4.84984752778921063201662023208, −4.70655597383091311843768232663, −4.38674743721012348506771336811, −4.10979918409140993415180965066, −3.99596295961969125037005694598, −3.68592502271996278038479052011, −3.16652957542397117331543919780, −3.09656774780074347514882328154, −3.03129174503983615466834982550, −2.53712261437844918334914973717, −2.46255843408285878057994504854, −2.25837556391202282049489462434, −1.55905565398484583550640728166, −1.41367915655116801823959000825, −1.34257307899782940176088316468, 0, 0, 0, 1.34257307899782940176088316468, 1.41367915655116801823959000825, 1.55905565398484583550640728166, 2.25837556391202282049489462434, 2.46255843408285878057994504854, 2.53712261437844918334914973717, 3.03129174503983615466834982550, 3.09656774780074347514882328154, 3.16652957542397117331543919780, 3.68592502271996278038479052011, 3.99596295961969125037005694598, 4.10979918409140993415180965066, 4.38674743721012348506771336811, 4.70655597383091311843768232663, 4.84984752778921063201662023208, 5.19512525411581451122285830797, 5.25197185070482291066827672635, 5.50420412663478528617957155201, 5.90693115706123916205285929007, 5.93510110653598198001483741427, 6.04819064830567895615015633523, 6.64437403536975662372635464262, 6.68863249250605218438765759024, 6.80446632040736850083736512314, 7.08934295705747738318694380838

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