Properties

Degree 8
Conductor $ 13^{4} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s − 5·3-s + 18·4-s − 30·5-s − 25·6-s − 15·7-s + 65·8-s + 22·9-s − 150·10-s − 17·11-s − 90·12-s + 125·13-s − 75·14-s + 150·15-s + 189·16-s − 70·17-s + 110·18-s + 141·19-s − 540·20-s + 75·21-s − 85·22-s − 145·23-s − 325·24-s + 275·25-s + 625·26-s + 65·27-s − 270·28-s + ⋯
L(s)  = 1  + 1.76·2-s − 0.962·3-s + 9/4·4-s − 2.68·5-s − 1.70·6-s − 0.809·7-s + 2.87·8-s + 0.814·9-s − 4.74·10-s − 0.465·11-s − 2.16·12-s + 2.66·13-s − 1.43·14-s + 2.58·15-s + 2.95·16-s − 0.998·17-s + 1.44·18-s + 1.70·19-s − 6.03·20-s + 0.779·21-s − 0.823·22-s − 1.31·23-s − 2.76·24-s + 11/5·25-s + 4.71·26-s + 0.463·27-s − 1.82·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(28561\)    =    \(13^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{13} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 28561,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $1.14691$
$L(\frac12)$  $\approx$  $1.14691$
$L(\frac{5}{2})$   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$C_2^2$ \( 1 - 125 T + 612 p T^{2} - 125 p^{3} T^{3} + p^{6} T^{4} \)
good2$D_4\times C_2$ \( 1 - 5 T + 7 T^{2} - 5 p T^{3} + 15 p^{2} T^{4} - 5 p^{4} T^{5} + 7 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \)
3$D_4\times C_2$ \( 1 + 5 T + p T^{2} - 160 T^{3} - 920 T^{4} - 160 p^{3} T^{5} + p^{7} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2^2$ \( ( 1 + 3 p T + 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 + 15 T - 513 T^{2} + 780 T^{3} + 349820 T^{4} + 780 p^{3} T^{5} - 513 p^{6} T^{6} + 15 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 17 T - 1489 T^{2} - 15028 T^{3} + 1005064 T^{4} - 15028 p^{3} T^{5} - 1489 p^{6} T^{6} + 17 p^{9} T^{7} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 + 70 T - 5063 T^{2} + 9590 T^{3} + 51050100 T^{4} + 9590 p^{3} T^{5} - 5063 p^{6} T^{6} + 70 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 141 T + 1299 T^{2} - 36096 p T^{3} + 448424 p^{2} T^{4} - 36096 p^{4} T^{5} + 1299 p^{6} T^{6} - 141 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 145 T - 3937 T^{2} + 91060 T^{3} + 219254380 T^{4} + 91060 p^{3} T^{5} - 3937 p^{6} T^{6} + 145 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 34 T - 32611 T^{2} - 510374 T^{3} + 517193284 T^{4} - 510374 p^{3} T^{5} - 32611 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 + 140 T + 21982 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 190 T - 66003 T^{2} - 151430 T^{3} + 6030722900 T^{4} - 151430 p^{3} T^{5} - 66003 p^{6} T^{6} - 190 p^{9} T^{7} + p^{12} T^{8} \)
41$D_4\times C_2$ \( 1 + 538 T + 80941 T^{2} + 38015618 T^{3} + 18774626844 T^{4} + 38015618 p^{3} T^{5} + 80941 p^{6} T^{6} + 538 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 + 455 T + 36243 T^{2} + 5354440 T^{3} + 6385191800 T^{4} + 5354440 p^{3} T^{5} + 36243 p^{6} T^{6} + 455 p^{9} T^{7} + p^{12} T^{8} \)
47$D_{4}$ \( ( 1 - 60 T + 125246 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 545 T + 256304 T^{2} - 545 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 809 T + 92959 T^{2} - 121968076 T^{3} + 138709769544 T^{4} - 121968076 p^{3} T^{5} + 92959 p^{6} T^{6} - 809 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 502 T - 94959 T^{2} - 53713498 T^{3} + 11662829084 T^{4} - 53713498 p^{3} T^{5} - 94959 p^{6} T^{6} + 502 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 475 T - 397113 T^{2} - 10075700 T^{3} + 229484582600 T^{4} - 10075700 p^{3} T^{5} - 397113 p^{6} T^{6} - 475 p^{9} T^{7} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 + 127 T - 656869 T^{2} - 5438648 T^{3} + 319053277564 T^{4} - 5438648 p^{3} T^{5} - 656869 p^{6} T^{6} + 127 p^{9} T^{7} + p^{12} T^{8} \)
73$D_{4}$ \( ( 1 - 585 T + 832884 T^{2} - 585 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 240 T + 993678 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 260 T + 1117974 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 921 T - 707351 T^{2} + 134147334 T^{3} + 1324901569974 T^{4} + 134147334 p^{3} T^{5} - 707351 p^{6} T^{6} + 921 p^{9} T^{7} + p^{12} T^{8} \)
97$D_4\times C_2$ \( 1 - 415 T - 761343 T^{2} + 370087870 T^{3} - 118607901730 T^{4} + 370087870 p^{3} T^{5} - 761343 p^{6} T^{6} - 415 p^{9} T^{7} + p^{12} T^{8} \)
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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

−15.09259625175710776437642716566, −14.55847462133591598399845749321, −13.74681360234623211552995837401, −13.55961242981874110454866407054, −13.37139333206078506665771272037, −13.14982582062895793753971993219, −12.24656201119802978584149472660, −12.14658068558979807099510010550, −11.85655960895952525637217088232, −11.56521870450852786934786447720, −11.12718488914698104803801456386, −10.80163656029455800316775363706, −10.56005695066024981131704880800, −9.876914585513506265401796361910, −9.009732338948342568151131608751, −8.162302933922337311779364217199, −8.137050051906496283020045415523, −7.34663469630280678944139989589, −6.90403086611684217568404125648, −6.58529346849352433488911864635, −5.64843544798853391137423986062, −5.36503335546656995246130403068, −4.19022483144951431422246083571, −3.83978155581531791135561886183, −3.58625571942394858965526178874, 3.58625571942394858965526178874, 3.83978155581531791135561886183, 4.19022483144951431422246083571, 5.36503335546656995246130403068, 5.64843544798853391137423986062, 6.58529346849352433488911864635, 6.90403086611684217568404125648, 7.34663469630280678944139989589, 8.137050051906496283020045415523, 8.162302933922337311779364217199, 9.009732338948342568151131608751, 9.876914585513506265401796361910, 10.56005695066024981131704880800, 10.80163656029455800316775363706, 11.12718488914698104803801456386, 11.56521870450852786934786447720, 11.85655960895952525637217088232, 12.14658068558979807099510010550, 12.24656201119802978584149472660, 13.14982582062895793753971993219, 13.37139333206078506665771272037, 13.55961242981874110454866407054, 13.74681360234623211552995837401, 14.55847462133591598399845749321, 15.09259625175710776437642716566

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