Properties

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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s − 10·8-s − 8·11-s + 4·13-s + 15·16-s − 2·17-s + 3·19-s + 24·22-s + 8·23-s − 12·26-s − 10·29-s − 21·32-s + 6·34-s + 4·37-s − 9·38-s − 4·41-s − 6·43-s − 48·44-s − 24·46-s − 8·47-s − 5·49-s + 24·52-s + 18·53-s + 30·58-s − 10·59-s + 14·61-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 3.53·8-s − 2.41·11-s + 1.10·13-s + 15/4·16-s − 0.485·17-s + 0.688·19-s + 5.11·22-s + 1.66·23-s − 2.35·26-s − 1.85·29-s − 3.71·32-s + 1.02·34-s + 0.657·37-s − 1.45·38-s − 0.624·41-s − 0.914·43-s − 7.23·44-s − 3.53·46-s − 1.16·47-s − 5/7·49-s + 3.32·52-s + 2.47·53-s + 3.93·58-s − 1.30·59-s + 1.79·61-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: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.288723764\)
\(L(\frac12)\) \(\approx\) \(1.288723764\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 8 T + 41 T^{2} + 160 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + 23 T^{2} - 72 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 31 T^{2} + 60 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 107 T^{2} + 572 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 4 T + 43 T^{2} - 72 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 5 T^{2} - 244 T^{3} + 5 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 8 T + 149 T^{2} + 736 T^{3} + 149 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 + 10 T + 173 T^{2} + 1172 T^{3} + 173 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 20 T + 249 T^{2} - 2360 T^{3} + 249 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 20 T + 261 T^{2} + 2520 T^{3} + 261 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 8 T + 155 T^{2} - 912 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 221 T^{2} - 16 T^{3} + 221 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 73 T^{2} - 504 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 24 T + 443 T^{2} + 4672 T^{3} + 443 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 22 T + 439 T^{2} - 4564 T^{3} + 439 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
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

−6.96868488315335213408946055544, −6.67199170811667325021736370539, −6.62487777698329249771059391991, −6.47324647198573705915022576547, −5.79047488542164358393696223561, −5.75643926846827802732727431159, −5.63019648636041995016179304690, −5.36046734424467050629469557726, −5.04839638533060583006422556797, −5.02521117219958121031058577034, −4.48585639664261130236534176268, −4.25368753709149120197544486505, −3.90853117734720178101251036931, −3.46286787526812040789781795366, −3.35943424569274292192689227386, −3.11902141056990383847453744543, −2.76576253754938955573947207686, −2.56409794199333565931121708535, −2.35683643481579403163961841076, −1.79581902305663366305092530818, −1.65570175032395001083512406353, −1.56373350943480798128875077943, −0.77727258028420918011490586216, −0.53372213282083449528426349315, −0.44685535770629483631864430847, 0.44685535770629483631864430847, 0.53372213282083449528426349315, 0.77727258028420918011490586216, 1.56373350943480798128875077943, 1.65570175032395001083512406353, 1.79581902305663366305092530818, 2.35683643481579403163961841076, 2.56409794199333565931121708535, 2.76576253754938955573947207686, 3.11902141056990383847453744543, 3.35943424569274292192689227386, 3.46286787526812040789781795366, 3.90853117734720178101251036931, 4.25368753709149120197544486505, 4.48585639664261130236534176268, 5.02521117219958121031058577034, 5.04839638533060583006422556797, 5.36046734424467050629469557726, 5.63019648636041995016179304690, 5.75643926846827802732727431159, 5.79047488542164358393696223561, 6.47324647198573705915022576547, 6.62487777698329249771059391991, 6.67199170811667325021736370539, 6.96868488315335213408946055544

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