Properties

Label 6-8550e3-1.1-c1e3-0-15
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s − 4·7-s + 10·8-s − 8·13-s − 12·14-s + 15·16-s − 2·17-s + 3·19-s − 24·26-s − 24·28-s + 8·29-s + 4·31-s + 21·32-s − 6·34-s − 14·37-s + 9·38-s − 2·41-s − 18·43-s − 14·47-s − 4·49-s − 48·52-s + 16·53-s − 40·56-s + 24·58-s − 2·59-s − 30·61-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.51·7-s + 3.53·8-s − 2.21·13-s − 3.20·14-s + 15/4·16-s − 0.485·17-s + 0.688·19-s − 4.70·26-s − 4.53·28-s + 1.48·29-s + 0.718·31-s + 3.71·32-s − 1.02·34-s − 2.30·37-s + 1.45·38-s − 0.312·41-s − 2.74·43-s − 2.04·47-s − 4/7·49-s − 6.65·52-s + 2.19·53-s − 5.34·56-s + 3.15·58-s − 0.260·59-s − 3.84·61-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$S_4\times C_2$ \( 1 + 4 T + 20 T^{2} + 54 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 23 T^{2} + 8 T^{3} + 23 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 8 T + 4 p T^{2} + 206 T^{3} + 4 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 44 T^{2} + 64 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 20 T^{2} + 122 T^{3} + 20 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 8 T + 36 T^{2} - 54 T^{3} + 36 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 4 T + p T^{2} - 16 T^{3} + p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 14 T + 151 T^{2} + 1052 T^{3} + 151 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 73 T^{2} + 264 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 18 T + 227 T^{2} + 1696 T^{3} + 227 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 14 T + 173 T^{2} + 1252 T^{3} + 173 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 16 T + 236 T^{2} - 34 p T^{3} + 236 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 2 T + 148 T^{2} + 156 T^{3} + 148 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 30 T + 473 T^{2} + 4552 T^{3} + 473 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 2 T + 140 T^{2} + 204 T^{3} + 140 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 91 T^{2} - 120 T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 124 T^{2} + 1624 T^{3} + 124 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T^{2} - 880 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 221 T^{2} + 988 T^{3} + 221 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 313 T^{2} - 2472 T^{3} + 313 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 10 T + 191 T^{2} + 1452 T^{3} + 191 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.06385191199383989682600033643, −6.77771724673184779813085565349, −6.61101785681870698963705330502, −6.58321268656818751121190727066, −6.30413941851520337767748445547, −6.14963542833399030206752401885, −5.83959579397251113365165470192, −5.34641373621416678235544347356, −5.20645198306941852159809004992, −5.16938426109606221631984888405, −4.78717664739826437236604103601, −4.73844675697588383476504646858, −4.59763757447936182882148379644, −4.12601504695656701092241953371, −3.73740995726483984368107024125, −3.67181168565848924557737223968, −3.25335353565895194655848022040, −3.22087620239924648564719942558, −2.97231525182228017746446668479, −2.49968479049327546957286625199, −2.41060726606394308702123904767, −2.38328540531084406620780391408, −1.47894176251802613675484926173, −1.43506477139600037136920619043, −1.33756216443445124111783273598, 0, 0, 0, 1.33756216443445124111783273598, 1.43506477139600037136920619043, 1.47894176251802613675484926173, 2.38328540531084406620780391408, 2.41060726606394308702123904767, 2.49968479049327546957286625199, 2.97231525182228017746446668479, 3.22087620239924648564719942558, 3.25335353565895194655848022040, 3.67181168565848924557737223968, 3.73740995726483984368107024125, 4.12601504695656701092241953371, 4.59763757447936182882148379644, 4.73844675697588383476504646858, 4.78717664739826437236604103601, 5.16938426109606221631984888405, 5.20645198306941852159809004992, 5.34641373621416678235544347356, 5.83959579397251113365165470192, 6.14963542833399030206752401885, 6.30413941851520337767748445547, 6.58321268656818751121190727066, 6.61101785681870698963705330502, 6.77771724673184779813085565349, 7.06385191199383989682600033643

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