Properties

Degree 8
Conductor $ 3^{8} \cdot 5^{4} \cdot 89^{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-s − 2·4-s + 4·5-s + 2·7-s + 6·8-s − 4·10-s + 14·11-s − 5·13-s − 2·14-s − 4·16-s + 3·17-s − 19-s − 8·20-s − 14·22-s + 3·23-s + 10·25-s + 5·26-s − 4·28-s + 10·29-s − 11·31-s − 11·32-s − 3·34-s + 8·35-s + 3·37-s + 38-s + 24·40-s + 3·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s + 1.78·5-s + 0.755·7-s + 2.12·8-s − 1.26·10-s + 4.22·11-s − 1.38·13-s − 0.534·14-s − 16-s + 0.727·17-s − 0.229·19-s − 1.78·20-s − 2.98·22-s + 0.625·23-s + 2·25-s + 0.980·26-s − 0.755·28-s + 1.85·29-s − 1.97·31-s − 1.94·32-s − 0.514·34-s + 1.35·35-s + 0.493·37-s + 0.162·38-s + 3.79·40-s + 0.468·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 89^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 89^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 5^{4} \cdot 89^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4005} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{8} \cdot 5^{4} \cdot 89^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $12.20323323$
$L(\frac12)$  $\approx$  $12.20323323$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{3,\;5,\;89\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
89$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + T + 3 T^{2} - T^{3} + 3 T^{4} - p T^{5} + 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 16 T^{2} - 24 T^{3} + 137 T^{4} - 24 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 114 T^{2} - 609 T^{3} + 2375 T^{4} - 609 p T^{5} + 114 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 38 T^{2} + 185 T^{3} + 669 T^{4} + 185 p T^{5} + 38 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 33 T^{2} - 154 T^{3} + 683 T^{4} - 154 p T^{5} + 33 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + T + 28 T^{2} - 9 T^{3} + 533 T^{4} - 9 p T^{5} + 28 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 75 T^{2} - 200 T^{3} + 2423 T^{4} - 200 p T^{5} + 75 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 112 T^{2} - 25 p T^{3} + 4813 T^{4} - 25 p^{2} T^{5} + 112 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 100 T^{2} + 709 T^{3} + 4879 T^{4} + 709 p T^{5} + 100 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 83 T^{2} - 354 T^{3} + 3943 T^{4} - 354 p T^{5} + 83 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 107 T^{2} - 512 T^{3} + 5333 T^{4} - 512 p T^{5} + 107 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 154 T^{2} - 1067 T^{3} + 9527 T^{4} - 1067 p T^{5} + 154 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 360 T^{2} - 3655 T^{3} + 28913 T^{4} - 3655 p T^{5} + 360 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 110 T^{2} + 355 T^{3} + 6533 T^{4} + 355 p T^{5} + 110 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 384 T^{2} - 4256 T^{3} + 38601 T^{4} - 4256 p T^{5} + 384 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 43 T^{2} + 616 T^{3} + 4025 T^{4} + 616 p T^{5} + 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 269 T^{2} + 1738 T^{3} + 26979 T^{4} + 1738 p T^{5} + 269 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 326 T^{2} + 3152 T^{3} + 35431 T^{4} + 3152 p T^{5} + 326 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 185 T^{2} - 620 T^{3} + 16923 T^{4} - 620 p T^{5} + 185 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 27 T + 481 T^{2} + 5558 T^{3} + 55595 T^{4} + 5558 p T^{5} + 481 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 209 T^{2} + 1238 T^{3} + 24167 T^{4} + 1238 p T^{5} + 209 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 41 T + 920 T^{2} - 14165 T^{3} + 161493 T^{4} - 14165 p T^{5} + 920 p^{2} T^{6} - 41 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.98628287630979239134620388049, −5.81397996285198339730365471142, −5.55851865590585429212043660524, −5.35414523167311294253984538633, −5.28689867147609591346868374570, −4.74576967549638086628869883419, −4.72376620810705676125894119643, −4.60535526190545172852769565753, −4.48221434061278478404093098781, −4.19968384219659426395014150928, −4.09011661908282239401677456840, −3.74315434451885671100319853688, −3.70355642119206810020935302391, −3.28145456542307878505249676004, −3.11695519039676024657672729734, −2.75892051525672126965771585104, −2.44145063898060468471683883780, −2.24257428578013519286376568567, −1.85101691958033073978640539281, −1.68489229668470201805370798918, −1.65072544106115250859749613367, −1.33689810125117329232002439819, −0.856155226417646382151279930522, −0.74525662982879634769340630898, −0.66100454836990368912986947045, 0.66100454836990368912986947045, 0.74525662982879634769340630898, 0.856155226417646382151279930522, 1.33689810125117329232002439819, 1.65072544106115250859749613367, 1.68489229668470201805370798918, 1.85101691958033073978640539281, 2.24257428578013519286376568567, 2.44145063898060468471683883780, 2.75892051525672126965771585104, 3.11695519039676024657672729734, 3.28145456542307878505249676004, 3.70355642119206810020935302391, 3.74315434451885671100319853688, 4.09011661908282239401677456840, 4.19968384219659426395014150928, 4.48221434061278478404093098781, 4.60535526190545172852769565753, 4.72376620810705676125894119643, 4.74576967549638086628869883419, 5.28689867147609591346868374570, 5.35414523167311294253984538633, 5.55851865590585429212043660524, 5.81397996285198339730365471142, 5.98628287630979239134620388049

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