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Properties

Label 8-7872e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.840\times 10^{15}$
Sign $1$
Analytic cond. $1.56116\times 10^{7}$
Root an. cond. $7.92831$
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  + 4·3-s + 4·5-s + 6·7-s + 10·9-s + 3·11-s + 2·13-s + 16·15-s − 3·17-s + 24·21-s − 2·25-s + 20·27-s + 13·29-s + 9·31-s + 12·33-s + 24·35-s + 7·37-s + 8·39-s − 4·41-s − 5·43-s + 40·45-s + 7·47-s + 4·49-s − 12·51-s + 16·53-s + 12·55-s + 10·59-s + 7·61-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 2.26·7-s + 10/3·9-s + 0.904·11-s + 0.554·13-s + 4.13·15-s − 0.727·17-s + 5.23·21-s − 2/5·25-s + 3.84·27-s + 2.41·29-s + 1.61·31-s + 2.08·33-s + 4.05·35-s + 1.15·37-s + 1.28·39-s − 0.624·41-s − 0.762·43-s + 5.96·45-s + 1.02·47-s + 4/7·49-s − 1.68·51-s + 2.19·53-s + 1.61·55-s + 1.30·59-s + 0.896·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(1.56116\times 10^{7}\)
Root analytic conductor: \(7.92831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(94.24455568\)
\(L(\frac12)\) \(\approx\) \(94.24455568\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
41$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 4 T + 18 T^{2} - 2 p^{2} T^{3} + 126 T^{4} - 2 p^{3} T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 6 T + 32 T^{2} - 120 T^{3} + 346 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 3 T + 35 T^{2} - 86 T^{3} + 524 T^{4} - 86 p T^{5} + 35 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2 T + 44 T^{2} - 60 T^{3} + 798 T^{4} - 60 p T^{5} + 44 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 3 T + 53 T^{2} + 164 T^{3} + 1226 T^{4} + 164 p T^{5} + 53 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 44 T^{2} + 30 T^{3} + 1078 T^{4} + 30 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 54 T^{2} + 62 T^{3} + 1514 T^{4} + 62 p T^{5} + 54 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 13 T + 129 T^{2} - 982 T^{3} + 5924 T^{4} - 982 p T^{5} + 129 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 9 T + 79 T^{2} - 200 T^{3} + 1424 T^{4} - 200 p T^{5} + 79 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 7 T + 85 T^{2} - 474 T^{3} + 4346 T^{4} - 474 p T^{5} + 85 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 5 T + 143 T^{2} + 660 T^{3} + 8604 T^{4} + 660 p T^{5} + 143 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 7 T + 175 T^{2} - 812 T^{3} + 11730 T^{4} - 812 p T^{5} + 175 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 16 T + 252 T^{2} - 2308 T^{3} + 20486 T^{4} - 2308 p T^{5} + 252 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 10 T + 224 T^{2} - 1554 T^{3} + 19118 T^{4} - 1554 p T^{5} + 224 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 7 T + 173 T^{2} - 550 T^{3} + 12042 T^{4} - 550 p T^{5} + 173 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 4 T + 156 T^{2} + 740 T^{3} + 12438 T^{4} + 740 p T^{5} + 156 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 13 T + 227 T^{2} - 2296 T^{3} + 21910 T^{4} - 2296 p T^{5} + 227 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 3 T + 193 T^{2} + 490 T^{3} + 18086 T^{4} + 490 p T^{5} + 193 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 6 T + 276 T^{2} - 1326 T^{3} + 31158 T^{4} - 1326 p T^{5} + 276 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 14 T + 110 T^{2} - 52 T^{3} - 566 T^{4} - 52 p T^{5} + 110 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 12 T + 116 T^{2} - 812 T^{3} - 8842 T^{4} - 812 p T^{5} + 116 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 10 T + 284 T^{2} + 2392 T^{3} + 39690 T^{4} + 2392 p T^{5} + 284 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.55101240663520272754456819826, −5.11520461359200199264649280432, −5.08899048061022708611556058181, −5.02998726719664237702457066609, −4.66968269518046321504557227892, −4.33695372533029423471335884561, −4.33371690266241531971845103646, −4.33187063550630213273475003850, −4.25124851875348587508762335112, −3.61452497979886437478923581794, −3.59588213179855562920868302323, −3.39668066194845775329190088541, −3.29632095990204598448716613595, −2.86244885886760712163435155547, −2.62705187327772141206296280632, −2.56256987547788107657143575741, −2.29277352019271772886040806315, −2.02304836625750625182883564326, −1.94237081801634655694226726519, −1.82824188785984469954460406248, −1.68806939548482653892132242532, −1.23125135515477936530931044396, −0.991541331946392411306464868488, −0.803105678056348789527314738930, −0.66096426252783768744892516083, 0.66096426252783768744892516083, 0.803105678056348789527314738930, 0.991541331946392411306464868488, 1.23125135515477936530931044396, 1.68806939548482653892132242532, 1.82824188785984469954460406248, 1.94237081801634655694226726519, 2.02304836625750625182883564326, 2.29277352019271772886040806315, 2.56256987547788107657143575741, 2.62705187327772141206296280632, 2.86244885886760712163435155547, 3.29632095990204598448716613595, 3.39668066194845775329190088541, 3.59588213179855562920868302323, 3.61452497979886437478923581794, 4.25124851875348587508762335112, 4.33187063550630213273475003850, 4.33371690266241531971845103646, 4.33695372533029423471335884561, 4.66968269518046321504557227892, 5.02998726719664237702457066609, 5.08899048061022708611556058181, 5.11520461359200199264649280432, 5.55101240663520272754456819826

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