Properties

Degree 8
Conductor $ 13^{4} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 8·4-s + 8·5-s + 16·6-s − 12·7-s − 4·8-s − 6·9-s − 32·10-s − 4·11-s − 32·12-s + 8·13-s + 48·14-s − 32·15-s − 25·16-s + 24·18-s + 64·20-s + 48·21-s + 16·22-s + 16·24-s + 32·25-s − 32·26-s + 40·27-s − 96·28-s + 40·29-s + 128·30-s + 40·31-s + ⋯
L(s)  = 1  − 2·2-s − 4/3·3-s + 2·4-s + 8/5·5-s + 8/3·6-s − 1.71·7-s − 1/2·8-s − 2/3·9-s − 3.19·10-s − 0.363·11-s − 8/3·12-s + 8/13·13-s + 24/7·14-s − 2.13·15-s − 1.56·16-s + 4/3·18-s + 16/5·20-s + 16/7·21-s + 8/11·22-s + 2/3·24-s + 1.27·25-s − 1.23·26-s + 1.48·27-s − 3.42·28-s + 1.37·29-s + 4.26·30-s + 1.29·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(28561\)    =    \(13^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{13} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 28561,\ (\ :1, 1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.109534$
$L(\frac12)$  $\approx$  $0.109534$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 13$, \(F_p\) is a polynomial of degree 8. If $p = 13$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad13$C_2^2$ \( 1 - 8 T + 8 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \)
good2$D_4\times C_2$ \( 1 + p^{2} T + p^{3} T^{2} + p^{2} T^{3} - 7 T^{4} + p^{4} T^{5} + p^{7} T^{6} + p^{8} T^{7} + p^{8} T^{8} \)
3$D_{4}$ \( ( 1 + 2 T + p^{2} T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 224 T^{3} + 1559 T^{4} - 224 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 744 T^{3} + 7519 T^{4} + 744 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 172 T^{3} - 2386 T^{4} + 172 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 418 T^{2} + 197763 T^{4} - 418 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^3$ \( 1 + 232162 T^{4} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 56 p T^{2} + 857778 T^{4} - 56 p^{5} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 20 T + 1532 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 40 T + 800 T^{2} - 39240 T^{3} + 1924322 T^{4} - 39240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 40 T + 800 T^{2} - 46560 T^{3} + 2667767 T^{4} - 46560 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 32 T + 512 T^{2} - 57248 T^{3} + 6389378 T^{4} - 57248 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \)
43$D_4\times C_2$ \( 1 - 6334 T^{2} + 16708731 T^{4} - 6334 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} - 1736 T^{3} - 6608737 T^{4} - 1736 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 40 T + 5768 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 56 T + 1568 T^{2} + 2632 T^{3} - 12442366 T^{4} + 2632 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 148 T + 12878 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 84 T + 3528 T^{2} - 155316 T^{3} - 33332642 T^{4} - 155316 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 4 p T + 8 p^{2} T^{2} - 57112 p T^{3} + 322400399 T^{4} - 57112 p^{3} T^{5} + 8 p^{6} T^{6} - 4 p^{7} T^{7} + p^{8} T^{8} \)
73$C_2^3$ \( 1 + 18164482 T^{4} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 - 32 T + 11528 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 52 T + 1352 T^{2} + 271804 T^{3} + 51880814 T^{4} + 271804 p^{2} T^{5} + 1352 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 200 T + 20000 T^{2} - 1908200 T^{3} + 179436962 T^{4} - 1908200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 68 T + 2312 T^{2} - 801924 T^{3} - 171375106 T^{4} - 801924 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \)
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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

−15.73267954907706978465117940994, −14.90831937210881891886498767921, −14.45109489363855657383888504907, −13.69557914963411258049827622329, −13.67826964445310405211522983516, −13.56268633963231116245228884102, −12.73477724760375199072324025869, −12.48767843173433937554344724520, −12.03463667937509193806138129804, −11.43954033197981425354098322200, −10.99395777799112031275258853299, −10.65636187705288183345272844949, −10.50876836275333162196408802027, −9.896974242426991948391952809843, −9.563996724071608323299871578343, −9.177307123611610263205322290920, −8.998398593171744386351339134601, −8.056957803555827474452859535269, −7.921204180954394322383080744709, −6.85874272103586218025205582972, −6.38261627920717730506117029383, −6.16404177135625374340439059988, −5.65990377670370487912622172668, −4.77450417013775926355845063463, −2.85218477777242428197479268848, 2.85218477777242428197479268848, 4.77450417013775926355845063463, 5.65990377670370487912622172668, 6.16404177135625374340439059988, 6.38261627920717730506117029383, 6.85874272103586218025205582972, 7.921204180954394322383080744709, 8.056957803555827474452859535269, 8.998398593171744386351339134601, 9.177307123611610263205322290920, 9.563996724071608323299871578343, 9.896974242426991948391952809843, 10.50876836275333162196408802027, 10.65636187705288183345272844949, 10.99395777799112031275258853299, 11.43954033197981425354098322200, 12.03463667937509193806138129804, 12.48767843173433937554344724520, 12.73477724760375199072324025869, 13.56268633963231116245228884102, 13.67826964445310405211522983516, 13.69557914963411258049827622329, 14.45109489363855657383888504907, 14.90831937210881891886498767921, 15.73267954907706978465117940994

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