Properties

Label 6-6930e3-1.1-c1e3-0-1
Degree $6$
Conductor $332812557000$
Sign $1$
Analytic cond. $169445.$
Root an. cond. $7.43883$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s − 3·5-s − 3·7-s − 10·8-s + 9·10-s − 3·11-s + 2·13-s + 9·14-s + 15·16-s − 2·17-s − 6·19-s − 18·20-s + 9·22-s − 10·23-s + 6·25-s − 6·26-s − 18·28-s − 12·31-s − 21·32-s + 6·34-s + 9·35-s + 4·37-s + 18·38-s + 30·40-s − 4·41-s + 8·43-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 1.34·5-s − 1.13·7-s − 3.53·8-s + 2.84·10-s − 0.904·11-s + 0.554·13-s + 2.40·14-s + 15/4·16-s − 0.485·17-s − 1.37·19-s − 4.02·20-s + 1.91·22-s − 2.08·23-s + 6/5·25-s − 1.17·26-s − 3.40·28-s − 2.15·31-s − 3.71·32-s + 1.02·34-s + 1.52·35-s + 0.657·37-s + 2.91·38-s + 4.74·40-s − 0.624·41-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(169445.\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3823444779\)
\(L(\frac12)\) \(\approx\) \(0.3823444779\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
11$C_1$ \( ( 1 + T )^{3} \)
good13$S_4\times C_2$ \( 1 - 2 T + 23 T^{2} - 36 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 23 T^{2} + 76 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 11 T^{2} - 4 T^{3} + 11 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 95 T^{2} + 476 T^{3} + 95 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 21 T^{2} + 92 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 12 T + 113 T^{2} + 712 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 109 T^{2} - 292 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 4 T + 33 T^{2} + 12 p T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 8 T + 21 T^{2} + 160 T^{3} + 21 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 29 T^{2} + 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 12 T + 141 T^{2} + 980 T^{3} + 141 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 4 T + 113 T^{2} - 344 T^{3} + 113 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 83 T^{2} + 388 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 4 T + 89 T^{2} - 600 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 28 T + 413 T^{2} - 4040 T^{3} + 413 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 14 T + 255 T^{2} + 2036 T^{3} + 255 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 18 T + 287 T^{2} - 2588 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 241 T^{2} - 1296 T^{3} + 241 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 18 T + 239 T^{2} + 1996 T^{3} + 239 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 137 T^{2} + 1092 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.34053806389158869371996173924, −6.67107685722981853069573175068, −6.66462126047522891641920631926, −6.62899616370167223782100440452, −6.07081673255192813437186709971, −6.04801143499359062044868100117, −5.98922604176304663676044977178, −5.28383837323806687741950422588, −5.23430529389800706436179969432, −5.10296802760846495978631745106, −4.38234936626767972937740745985, −4.21641061333996531826932461763, −4.09078616397817139346086260539, −3.72237190359251210183024124206, −3.43264114714875048809737981738, −3.41034116126752265772606894938, −2.78706925799825349980546117231, −2.65575464892903192954408223640, −2.42548850872596634164798608899, −1.90798638556792342369918824206, −1.76441068357714542664903479573, −1.56751638773802829439853075035, −0.66264159211278378958030835745, −0.44289500760624363717499421018, −0.36701066668919928526512795685, 0.36701066668919928526512795685, 0.44289500760624363717499421018, 0.66264159211278378958030835745, 1.56751638773802829439853075035, 1.76441068357714542664903479573, 1.90798638556792342369918824206, 2.42548850872596634164798608899, 2.65575464892903192954408223640, 2.78706925799825349980546117231, 3.41034116126752265772606894938, 3.43264114714875048809737981738, 3.72237190359251210183024124206, 4.09078616397817139346086260539, 4.21641061333996531826932461763, 4.38234936626767972937740745985, 5.10296802760846495978631745106, 5.23430529389800706436179969432, 5.28383837323806687741950422588, 5.98922604176304663676044977178, 6.04801143499359062044868100117, 6.07081673255192813437186709971, 6.62899616370167223782100440452, 6.66462126047522891641920631926, 6.67107685722981853069573175068, 7.34053806389158869371996173924

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