Properties

Degree 8
Conductor $ 2^{8} \cdot 41^{4} $
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  + 2·3-s + 4·5-s + 2·9-s + 4·11-s + 8·15-s − 4·17-s + 6·19-s − 12·23-s + 4·25-s − 4·27-s − 4·29-s − 8·31-s + 8·33-s + 16·37-s − 4·41-s + 4·43-s + 8·45-s − 6·47-s − 6·49-s − 8·51-s − 16·53-s + 16·55-s + 12·57-s + 12·59-s + 24·61-s + 28·67-s − 24·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 2/3·9-s + 1.20·11-s + 2.06·15-s − 0.970·17-s + 1.37·19-s − 2.50·23-s + 4/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s + 1.39·33-s + 2.63·37-s − 0.624·41-s + 0.609·43-s + 1.19·45-s − 0.875·47-s − 6/7·49-s − 1.12·51-s − 2.19·53-s + 2.15·55-s + 1.58·57-s + 1.56·59-s + 3.07·61-s + 3.42·67-s − 2.88·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 41^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{164} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $2.84883$
$L(\frac12)$  $\approx$  $2.84883$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;41\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
41$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 2 T + 2 T^{2} + 4 T^{3} - 8 T^{4} + 4 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 4 T + 12 T^{2} - 16 T^{3} + 34 T^{4} - 16 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 6 T^{2} + 26 T^{3} + 24 T^{4} + 26 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 114 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 12 T^{2} - 48 T^{3} + 118 T^{4} - 48 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 4 T + 20 T^{2} + 124 T^{3} + 534 T^{4} + 124 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 6 T + 62 T^{2} - 208 T^{3} + 1448 T^{4} - 208 p T^{5} + 62 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 700 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 76 T^{2} + 12 p T^{3} + 2870 T^{4} + 12 p^{2} T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 712 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 1740 p T^{5} + 212 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 4 T + 124 T^{2} - 244 T^{3} + 6678 T^{4} - 244 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 6 T + 126 T^{2} + 640 T^{3} + 8608 T^{4} + 640 p T^{5} + 126 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 16 T + 4 p T^{2} + 1824 T^{3} + 15558 T^{4} + 1824 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 1996 p T^{5} + 252 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 4824 p T^{5} + 420 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 28 T + 538 T^{2} - 6638 T^{3} + 64208 T^{4} - 6638 p T^{5} + 538 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 1060 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 4224 p T^{5} + 366 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 12 T + 252 T^{2} + 1644 T^{3} + 24598 T^{4} + 1644 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 4 T + 228 T^{2} - 796 T^{3} + 326 p T^{4} - 796 p T^{5} + 228 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 16 T + 268 T^{2} - 3376 T^{3} + 38118 T^{4} - 3376 p T^{5} + 268 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−9.431271901955034512661160144361, −9.257187464030829018933032864878, −9.113520033222656878036949681569, −8.564538758740434850208090529504, −8.410814602927378954435140271782, −8.075933363635434941303925882581, −7.83558647351392841848529880578, −7.67071754643379739839855280267, −7.09038681971967530591367393449, −6.99601981099149970700711301452, −6.48411868103710289674271580053, −6.28520117616281367417948085015, −6.16520671841520752862165626534, −5.53574675057552714505672444117, −5.41014431862081206521195436965, −5.35355840171940356639548895826, −4.55854470282430128721053511947, −4.17490289254241671008576463925, −3.80460196249842388129974887510, −3.75351568794207299543189756172, −3.17236024489173932430653019754, −2.46650905065687307909939729608, −2.23649406422758071343303763367, −1.90041449727506725045277223184, −1.39280173733790821584173906338, 1.39280173733790821584173906338, 1.90041449727506725045277223184, 2.23649406422758071343303763367, 2.46650905065687307909939729608, 3.17236024489173932430653019754, 3.75351568794207299543189756172, 3.80460196249842388129974887510, 4.17490289254241671008576463925, 4.55854470282430128721053511947, 5.35355840171940356639548895826, 5.41014431862081206521195436965, 5.53574675057552714505672444117, 6.16520671841520752862165626534, 6.28520117616281367417948085015, 6.48411868103710289674271580053, 6.99601981099149970700711301452, 7.09038681971967530591367393449, 7.67071754643379739839855280267, 7.83558647351392841848529880578, 8.075933363635434941303925882581, 8.410814602927378954435140271782, 8.564538758740434850208090529504, 9.113520033222656878036949681569, 9.257187464030829018933032864878, 9.431271901955034512661160144361

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