Properties

Degree $8$
Conductor $3111696$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 8·7-s − 6·9-s − 24·11-s + 12·16-s + 24·23-s + 64·25-s + 32·28-s + 120·29-s − 24·36-s − 80·37-s − 128·43-s − 96·44-s + 22·49-s − 216·53-s − 48·63-s + 32·64-s + 176·67-s − 120·71-s − 192·77-s + 128·79-s + 27·81-s + 96·92-s + 144·99-s + 256·100-s − 168·107-s − 8·109-s + ⋯
L(s)  = 1  + 4-s + 8/7·7-s − 2/3·9-s − 2.18·11-s + 3/4·16-s + 1.04·23-s + 2.55·25-s + 8/7·28-s + 4.13·29-s − 2/3·36-s − 2.16·37-s − 2.97·43-s − 2.18·44-s + 0.448·49-s − 4.07·53-s − 0.761·63-s + 1/2·64-s + 2.62·67-s − 1.69·71-s − 2.49·77-s + 1.62·79-s + 1/3·81-s + 1.04·92-s + 1.45·99-s + 2.55·100-s − 1.57·107-s − 0.0733·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(2\)
Character: induced by $\chi_{42} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3111696,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.51163\)
\(L(\frac12)\) \(\approx\) \(1.51163\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 8 T + 6 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \)
good5$D_4\times C_2$ \( 1 - 64 T^{2} + 1986 T^{4} - 64 p^{4} T^{6} + p^{8} T^{8} \)
11$D_{4}$ \( ( 1 + 12 T + 260 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \)
17$C_2^3$ \( 1 + 320 T^{2} + 546 p^{2} T^{4} + 320 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 652 T^{2} + 348486 T^{4} - 652 p^{4} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 - 12 T + 932 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 - 1252 T^{2} + 745926 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 40 T + 546 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 64 T + 2922 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3940 T^{2} + 5766 p^{2} T^{4} - 3940 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 108 T + 8246 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 4420 T^{2} + 6539622 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 14596 T^{2} + 80934054 T^{4} - 14596 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 88 T + 10626 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 60 T + 4484 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 13108 T^{2} + 98972646 T^{4} - 13108 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 - 64 T + 3138 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 8896 T^{2} + 52680354 T^{4} - 8896 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.97128720903096030219212174477, −11.41357024915097376533190243079, −10.86519446483954773602597802227, −10.83891403603784171011298473820, −10.80692096612039851866663777859, −10.26337553007293711497230863901, −10.13599273426427390814173032677, −9.577375766818075752095347936206, −9.060027996856958197971697759825, −8.545921331359559039305387495455, −8.240003287991511385307968377459, −8.203509634533493905314742065359, −7.975142494930554989947163677204, −7.27641463192859787441316648427, −6.78491589255249120465243156852, −6.58411108149393790950686580729, −6.44305515098406358521201467802, −5.45319462803678424647293376295, −5.05233619897191696013507298883, −4.90431479939002232601911369007, −4.76563063945721265405188761708, −3.31428425515721586056396944495, −2.93119924689143621118693881153, −2.65424106596318642352946289616, −1.47387573767770102315147410442, 1.47387573767770102315147410442, 2.65424106596318642352946289616, 2.93119924689143621118693881153, 3.31428425515721586056396944495, 4.76563063945721265405188761708, 4.90431479939002232601911369007, 5.05233619897191696013507298883, 5.45319462803678424647293376295, 6.44305515098406358521201467802, 6.58411108149393790950686580729, 6.78491589255249120465243156852, 7.27641463192859787441316648427, 7.975142494930554989947163677204, 8.203509634533493905314742065359, 8.240003287991511385307968377459, 8.545921331359559039305387495455, 9.060027996856958197971697759825, 9.577375766818075752095347936206, 10.13599273426427390814173032677, 10.26337553007293711497230863901, 10.80692096612039851866663777859, 10.83891403603784171011298473820, 10.86519446483954773602597802227, 11.41357024915097376533190243079, 11.97128720903096030219212174477

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