Properties

Label 6-8470e3-1.1-c1e3-0-0
Degree $6$
Conductor $607645423000$
Sign $1$
Analytic cond. $309372.$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s − 3·5-s − 3·7-s − 10·8-s − 2·9-s + 9·10-s + 9·14-s + 15·16-s − 11·17-s + 6·18-s − 5·19-s − 18·20-s − 4·23-s + 6·25-s − 2·27-s − 18·28-s − 2·29-s − 21·32-s + 33·34-s + 9·35-s − 12·36-s − 2·37-s + 15·38-s + 30·40-s − 7·43-s + 6·45-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 1.34·5-s − 1.13·7-s − 3.53·8-s − 2/3·9-s + 2.84·10-s + 2.40·14-s + 15/4·16-s − 2.66·17-s + 1.41·18-s − 1.14·19-s − 4.02·20-s − 0.834·23-s + 6/5·25-s − 0.384·27-s − 3.40·28-s − 0.371·29-s − 3.71·32-s + 5.65·34-s + 1.52·35-s − 2·36-s − 0.328·37-s + 2.43·38-s + 4.74·40-s − 1.06·43-s + 0.894·45-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(309372.\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1559196010\)
\(L(\frac12)\) \(\approx\) \(0.1559196010\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 + 2 T^{2} + 2 T^{3} + 2 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 32 T^{2} - 2 T^{3} + 32 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 11 T + 75 T^{2} + 342 T^{3} + 75 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 5 T + 36 T^{2} + 173 T^{3} + 36 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 42 T^{2} + 78 T^{3} + 42 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 23 T^{2} + 244 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$C_2$ \( ( 1 + p T^{2} )^{3} \)
37$S_4\times C_2$ \( 1 + 2 T + 43 T^{2} + 140 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 95 T^{2} + 16 T^{3} + 95 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 7 T + 41 T^{2} + 54 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 93 T^{2} - 624 T^{3} + 93 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - T + 143 T^{2} - 122 T^{3} + 143 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 7 T + 164 T^{2} - 743 T^{3} + 164 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 13 T + 3 p T^{2} + 1270 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 81 T^{2} + 214 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 17 T + 277 T^{2} - 2478 T^{3} + 277 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 19 T + 307 T^{2} + 2758 T^{3} + 307 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 9 T + 236 T^{2} - 1381 T^{3} + 236 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 254 T^{2} + 988 T^{3} + 254 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 12 T + 287 T^{2} - 2104 T^{3} + 287 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 7 T + 275 T^{2} + 1230 T^{3} + 275 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.81321886567348394650292663318, −6.74190766998415562510693550660, −6.54720096294527884778896129189, −6.38566070546845016662251589396, −6.13878702629435627419495201504, −5.95230658654332882319476083218, −5.69628624011442515419634796357, −5.28671378447508943816599371892, −5.00069986476636588306134793337, −4.73320694631123256639922355464, −4.41931569672592094681686637236, −4.14009276836055026292857283710, −3.96580004903656535773754381262, −3.64464505615654968763636287049, −3.44366016535174085310943197547, −3.15864456810663634427561614614, −2.66316768444933657267058087177, −2.59788087532886143129095906625, −2.38618737615131979859249304919, −1.95477141460989191437762761555, −1.72665894949640718365850682916, −1.43212147698443262622523245458, −0.66673720919612698839397987376, −0.42290285968041852322034220504, −0.21893070082006326611047469333, 0.21893070082006326611047469333, 0.42290285968041852322034220504, 0.66673720919612698839397987376, 1.43212147698443262622523245458, 1.72665894949640718365850682916, 1.95477141460989191437762761555, 2.38618737615131979859249304919, 2.59788087532886143129095906625, 2.66316768444933657267058087177, 3.15864456810663634427561614614, 3.44366016535174085310943197547, 3.64464505615654968763636287049, 3.96580004903656535773754381262, 4.14009276836055026292857283710, 4.41931569672592094681686637236, 4.73320694631123256639922355464, 5.00069986476636588306134793337, 5.28671378447508943816599371892, 5.69628624011442515419634796357, 5.95230658654332882319476083218, 6.13878702629435627419495201504, 6.38566070546845016662251589396, 6.54720096294527884778896129189, 6.74190766998415562510693550660, 6.81321886567348394650292663318

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