Properties

Label 8-12e12-1.1-c2e4-0-9
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $4.91490\times 10^{6}$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s − 7-s − 36·11-s − 5·13-s − 2·19-s − 99·23-s + 22·25-s − 63·29-s − 7·31-s − 9·35-s + 64·37-s + 18·41-s + 46·43-s + 81·47-s + 24·49-s − 324·55-s + 126·59-s − 41·61-s − 45·65-s − 116·67-s + 86·73-s + 36·77-s + 83·79-s − 81·83-s + 5·91-s − 18·95-s − 196·97-s + ⋯
L(s)  = 1  + 9/5·5-s − 1/7·7-s − 3.27·11-s − 0.384·13-s − 0.105·19-s − 4.30·23-s + 0.879·25-s − 2.17·29-s − 0.225·31-s − 0.257·35-s + 1.72·37-s + 0.439·41-s + 1.06·43-s + 1.72·47-s + 0.489·49-s − 5.89·55-s + 2.13·59-s − 0.672·61-s − 0.692·65-s − 1.73·67-s + 1.17·73-s + 0.467·77-s + 1.05·79-s − 0.975·83-s + 5/91·91-s − 0.189·95-s − 2.02·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(4.91490\times 10^{6}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9423294934\)
\(L(\frac12)\) \(\approx\) \(0.9423294934\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 9 T + 59 T^{2} - 288 T^{3} + 1074 T^{4} - 288 p^{2} T^{5} + 59 p^{4} T^{6} - 9 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + T - 23 T^{2} - 74 T^{3} - 1874 T^{4} - 74 p^{2} T^{5} - 23 p^{4} T^{6} + p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 9036 p^{2} T^{5} + 683 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 + 5 T - 245 T^{2} - 340 T^{3} + 40114 T^{4} - 340 p^{2} T^{5} - 245 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 769 T^{2} + 298176 T^{4} - 769 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 + T + 648 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 183150 p^{2} T^{5} + 5117 p^{4} T^{6} + 99 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 50400 p^{2} T^{5} + 2123 p^{4} T^{6} + 63 p^{6} T^{7} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4592 p^{2} T^{5} - 1217 p^{4} T^{6} + 7 p^{6} T^{7} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 32 T + 1806 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 32490 p^{2} T^{5} + 1913 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2$ \( ( 1 - 23 T - 1320 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 384102 p^{2} T^{5} + 6929 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 7204 T^{2} + 26018214 T^{4} - 7204 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 844326 p^{2} T^{5} + 11993 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 + 41 T - 5513 T^{2} - 10168 T^{3} + 31652794 T^{4} - 10168 p^{2} T^{5} - 5513 p^{4} T^{6} + 41 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 80156 p^{2} T^{5} + 3787 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 43 T + 10452 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 11122 p^{2} T^{5} - 5459 p^{4} T^{6} - 83 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 + 81 T + 16289 T^{2} + 1142262 T^{3} + 166474326 T^{4} + 1142262 p^{2} T^{5} + 16289 p^{4} T^{6} + 81 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 1824172 p^{2} T^{5} + 10291 p^{4} T^{6} + 196 p^{6} T^{7} + p^{8} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.23102496539151679262892768679, −6.11213311510625734424354166258, −6.10473661985494391382441382310, −5.67215948941419450408283141530, −5.64425779641710426578729046164, −5.54779004165535702145141752503, −5.21208870703570201545509928836, −5.14507533088394652117015508120, −4.76857649865393333233312737094, −4.36501552705546187632614420598, −4.32215330470273092374758945743, −4.07121394086616238684426293341, −3.75386361707268597468417456784, −3.45060601415469129935581441389, −3.43450793690496794669187341385, −2.59204200890360450930315458894, −2.54180178686418006063698110219, −2.46108088740771787740585238906, −2.38863506490172296380219124004, −2.00147730258222819063180793169, −1.89068879137958828262916447284, −1.38622231767998278882000173551, −1.01125906820547450643800429045, −0.33563588511162485271406719555, −0.20760887160761254164935793345, 0.20760887160761254164935793345, 0.33563588511162485271406719555, 1.01125906820547450643800429045, 1.38622231767998278882000173551, 1.89068879137958828262916447284, 2.00147730258222819063180793169, 2.38863506490172296380219124004, 2.46108088740771787740585238906, 2.54180178686418006063698110219, 2.59204200890360450930315458894, 3.43450793690496794669187341385, 3.45060601415469129935581441389, 3.75386361707268597468417456784, 4.07121394086616238684426293341, 4.32215330470273092374758945743, 4.36501552705546187632614420598, 4.76857649865393333233312737094, 5.14507533088394652117015508120, 5.21208870703570201545509928836, 5.54779004165535702145141752503, 5.64425779641710426578729046164, 5.67215948941419450408283141530, 6.10473661985494391382441382310, 6.11213311510625734424354166258, 6.23102496539151679262892768679

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