Properties

Degree 8
Conductor $ 3^{8} \cdot 7^{4} \cdot 11^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·4-s + 4·5-s + 4·7-s − 6·8-s + 4·10-s − 6·13-s + 4·14-s − 4·16-s + 8·17-s + 10·19-s − 8·20-s + 10·23-s + 4·25-s − 6·26-s − 8·28-s − 18·31-s + 11·32-s + 8·34-s + 16·35-s − 2·37-s + 10·38-s − 24·40-s + 10·41-s + 4·43-s + 10·46-s − 4·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 1.78·5-s + 1.51·7-s − 2.12·8-s + 1.26·10-s − 1.66·13-s + 1.06·14-s − 16-s + 1.94·17-s + 2.29·19-s − 1.78·20-s + 2.08·23-s + 4/5·25-s − 1.17·26-s − 1.51·28-s − 3.23·31-s + 1.94·32-s + 1.37·34-s + 2.70·35-s − 0.328·37-s + 1.62·38-s − 3.79·40-s + 1.56·41-s + 0.609·43-s + 1.47·46-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $20.04654643$
$L(\frac12)$  $\approx$  $20.04654643$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - T + 3 T^{2} + T^{3} + 3 T^{4} + p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 44 T^{2} + 218 T^{3} + 822 T^{4} + 218 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 48 T^{2} - 264 T^{3} + 1358 T^{4} - 264 p T^{5} + 48 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 100 T^{2} - 30 p T^{3} + 3062 T^{4} - 30 p^{2} T^{5} + 100 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 83 T^{2} - 500 T^{3} + 2869 T^{4} - 500 p T^{5} + 83 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + 37 T^{2} + 17 p T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 192 T^{2} + 1362 T^{3} + 8238 T^{4} + 1362 p T^{5} + 192 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 71 T^{2} + 4 T^{3} + 2797 T^{4} + 4 p T^{5} + 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 60 T^{2} + 290 T^{3} - 2938 T^{4} + 290 p T^{5} + 60 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 69 T^{2} - 392 T^{3} + 4097 T^{4} - 392 p T^{5} + 69 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 108 T^{2} - 4 T^{3} + 4758 T^{4} - 4 p T^{5} + 108 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 161 T^{2} + 20 T^{3} + 11657 T^{4} + 20 p T^{5} + 161 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 308 T^{2} - 2936 T^{3} + 29398 T^{4} - 2936 p T^{5} + 308 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 260 T^{2} + 2306 T^{3} + 24294 T^{4} + 2306 p T^{5} + 260 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 6696 p T^{5} + 541 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 327 T^{2} - 3632 T^{3} + 36173 T^{4} - 3632 p T^{5} + 327 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 284 T^{2} - 852 T^{3} + 30822 T^{4} - 852 p T^{5} + 284 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 445 T^{2} - 5040 T^{3} + 57977 T^{4} - 5040 p T^{5} + 445 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 212 T^{2} - 982 T^{3} + 24278 T^{4} - 982 p T^{5} + 212 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 308 T^{2} - 3096 T^{3} + 38678 T^{4} - 3096 p T^{5} + 308 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 656 T^{2} - 8944 T^{3} + 99982 T^{4} - 8944 p T^{5} + 656 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.48068889902770749536312264589, −5.22525432472344898664076075978, −5.19200752088708261241976640210, −5.15627327738166369603904010060, −4.91082227435794543180069719380, −4.56239202243674935957118415622, −4.54496004859240786454342065767, −4.42636860036234035605597657011, −4.15956440382082611091685504089, −3.70066824518540539816468402810, −3.51425308300654555140814826766, −3.46184207845872961317683322444, −3.18864191924024847389062700062, −3.05474524299489968316174172664, −2.99214437919731132254633833063, −2.68089949213873496284019526136, −2.24605934580626206109942644491, −2.00519686638820252879694456824, −1.96946523251414126426304561584, −1.74898812712152215982925946153, −1.68504486501141907441597385027, −0.921675395505286575241811086068, −0.824686871640981484672174920591, −0.60466661847827539411606027653, −0.58434204653100018857957110606, 0.58434204653100018857957110606, 0.60466661847827539411606027653, 0.824686871640981484672174920591, 0.921675395505286575241811086068, 1.68504486501141907441597385027, 1.74898812712152215982925946153, 1.96946523251414126426304561584, 2.00519686638820252879694456824, 2.24605934580626206109942644491, 2.68089949213873496284019526136, 2.99214437919731132254633833063, 3.05474524299489968316174172664, 3.18864191924024847389062700062, 3.46184207845872961317683322444, 3.51425308300654555140814826766, 3.70066824518540539816468402810, 4.15956440382082611091685504089, 4.42636860036234035605597657011, 4.54496004859240786454342065767, 4.56239202243674935957118415622, 4.91082227435794543180069719380, 5.15627327738166369603904010060, 5.19200752088708261241976640210, 5.22525432472344898664076075978, 5.48068889902770749536312264589

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