Properties

Degree 8
Conductor $ 7^{4} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 8·3-s + 28·4-s + 38·5-s − 16·6-s − 168·7-s − 32·8-s + 465·9-s − 76·10-s − 424·11-s + 224·12-s − 1.84e3·13-s + 336·14-s + 304·15-s + 960·16-s + 2.34e3·17-s − 930·18-s + 360·19-s + 1.06e3·20-s − 1.34e3·21-s + 848·22-s + 12·23-s − 256·24-s + 2.91e3·25-s + 3.69e3·26-s + 6.76e3·27-s − 4.70e3·28-s + ⋯
L(s)  = 1  − 0.353·2-s + 0.513·3-s + 7/8·4-s + 0.679·5-s − 0.181·6-s − 1.29·7-s − 0.176·8-s + 1.91·9-s − 0.240·10-s − 1.05·11-s + 0.449·12-s − 3.03·13-s + 0.458·14-s + 0.348·15-s + 0.937·16-s + 1.96·17-s − 0.676·18-s + 0.228·19-s + 0.594·20-s − 0.665·21-s + 0.373·22-s + 0.00473·23-s − 0.0907·24-s + 0.931·25-s + 1.07·26-s + 1.78·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2401\)    =    \(7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{7} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2401,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)$
$L(3)$  $\approx$  $1.31535$
$L(\frac12)$  $\approx$  $1.31535$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 7$, \(F_p\) is a polynomial of degree 8. If $p = 7$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad7$C_2^2$ \( 1 + 24 p T + 34 p^{3} T^{2} + 24 p^{6} T^{3} + p^{10} T^{4} \)
good2$D_4\times C_2$ \( 1 + p T - 3 p^{3} T^{2} - 9 p^{3} T^{3} - 23 p^{4} T^{4} - 9 p^{8} T^{5} - 3 p^{13} T^{6} + p^{16} T^{7} + p^{20} T^{8} \)
3$D_4\times C_2$ \( 1 - 8 T - 401 T^{2} + 56 p T^{3} + 15736 p^{2} T^{4} + 56 p^{6} T^{5} - 401 p^{10} T^{6} - 8 p^{15} T^{7} + p^{20} T^{8} \)
5$D_4\times C_2$ \( 1 - 38 T - 1467 T^{2} + 126882 T^{3} - 5804204 T^{4} + 126882 p^{5} T^{5} - 1467 p^{10} T^{6} - 38 p^{15} T^{7} + p^{20} T^{8} \)
11$D_4\times C_2$ \( 1 + 424 T - 167697 T^{2} + 10757304 T^{3} + 65846956552 T^{4} + 10757304 p^{5} T^{5} - 167697 p^{10} T^{6} + 424 p^{15} T^{7} + p^{20} T^{8} \)
13$D_{4}$ \( ( 1 + 924 T + 927022 T^{2} + 924 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 138 p T + 1932761 T^{2} - 100911258 p T^{3} + 2921236022964 T^{4} - 100911258 p^{6} T^{5} + 1932761 p^{10} T^{6} - 138 p^{16} T^{7} + p^{20} T^{8} \)
19$D_4\times C_2$ \( 1 - 360 T - 2016025 T^{2} + 1010366280 T^{3} - 1848262047576 T^{4} + 1010366280 p^{5} T^{5} - 2016025 p^{10} T^{6} - 360 p^{15} T^{7} + p^{20} T^{8} \)
23$D_4\times C_2$ \( 1 - 12 T - 12696421 T^{2} + 2113452 T^{3} + 119775324752184 T^{4} + 2113452 p^{5} T^{5} - 12696421 p^{10} T^{6} - 12 p^{15} T^{7} + p^{20} T^{8} \)
29$D_{4}$ \( ( 1 + 7052 T + 35324974 T^{2} + 7052 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 3548 T - 28901749 T^{2} - 55945747452 T^{3} + 541403754912104 T^{4} - 55945747452 p^{5} T^{5} - 28901749 p^{10} T^{6} + 3548 p^{15} T^{7} + p^{20} T^{8} \)
37$D_4\times C_2$ \( 1 + 11090 T - 23355139 T^{2} + 84897554250 T^{3} + 8079277710681572 T^{4} + 84897554250 p^{5} T^{5} - 23355139 p^{10} T^{6} + 11090 p^{15} T^{7} + p^{20} T^{8} \)
41$D_{4}$ \( ( 1 - 3500 T + 206898214 T^{2} - 3500 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 12680 T + 267378054 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 22956 T + 35452763 T^{2} - 753763910004 T^{3} + 68138105036084328 T^{4} - 753763910004 p^{5} T^{5} + 35452763 p^{10} T^{6} - 22956 p^{15} T^{7} + p^{20} T^{8} \)
53$D_4\times C_2$ \( 1 + 3042 T - 707161075 T^{2} - 364967439174 T^{3} + 334492868775797220 T^{4} - 364967439174 p^{5} T^{5} - 707161075 p^{10} T^{6} + 3042 p^{15} T^{7} + p^{20} T^{8} \)
59$D_4\times C_2$ \( 1 - 65808 T + 1828603607 T^{2} - 70562013287472 T^{3} + 2653216336714668312 T^{4} - 70562013287472 p^{5} T^{5} + 1828603607 p^{10} T^{6} - 65808 p^{15} T^{7} + p^{20} T^{8} \)
61$D_4\times C_2$ \( 1 - 42486 T + 303064037 T^{2} + 7953228077298 T^{3} + 18102385363935684 T^{4} + 7953228077298 p^{5} T^{5} + 303064037 p^{10} T^{6} - 42486 p^{15} T^{7} + p^{20} T^{8} \)
67$D_4\times C_2$ \( 1 + 42312 T - 1326272449 T^{2} + 17615652522648 T^{3} + 5473083141138317592 T^{4} + 17615652522648 p^{5} T^{5} - 1326272449 p^{10} T^{6} + 42312 p^{15} T^{7} + p^{20} T^{8} \)
71$D_{4}$ \( ( 1 + 2208 T + 3433192846 T^{2} + 2208 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 50506 T - 222184951 T^{2} + 69349899662694 T^{3} - 1895976700185808828 T^{4} + 69349899662694 p^{5} T^{5} - 222184951 p^{10} T^{6} - 50506 p^{15} T^{7} + p^{20} T^{8} \)
79$D_4\times C_2$ \( 1 + 9004 T - 5095520573 T^{2} - 8801591961836 T^{3} + 17079371584250245336 T^{4} - 8801591961836 p^{5} T^{5} - 5095520573 p^{10} T^{6} + 9004 p^{15} T^{7} + p^{20} T^{8} \)
83$D_{4}$ \( ( 1 + 104328 T + 9837878230 T^{2} + 104328 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 26666 T - 7396026999 T^{2} + 81625061802438 T^{3} + 30572703729886615780 T^{4} + 81625061802438 p^{5} T^{5} - 7396026999 p^{10} T^{6} - 26666 p^{15} T^{7} + p^{20} T^{8} \)
97$D_{4}$ \( ( 1 - 2156 p T + 28107307478 T^{2} - 2156 p^{6} T^{3} + p^{10} T^{4} )^{2} \)
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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

−16.47192746263695183583764465299, −16.03801878611397430936717703502, −15.94156673157555611824397647322, −15.15826741594825591264719986097, −14.77321029738825350425838697111, −14.70834586642623482984680703603, −14.13425557388889850963185316499, −13.17891789008517935944615106046, −13.11981359757681676277581925418, −12.52412358353080258310017563007, −12.44501284768504795812830520917, −11.90327030397803109767832616100, −10.99748335226301824877763326540, −10.14284762309064775210492991809, −9.998802688611086109464520430262, −9.946388780896094143993725974526, −9.454139797789076397700060710186, −8.435416633558901220546111564259, −7.37732792484842868284117396695, −7.25821217549352394679105745722, −7.10655360964114683232862426342, −5.64713656488976520025843190553, −5.12553753782728504599967213955, −3.43551399195511712236903657816, −2.24580681910652224236344448604, 2.24580681910652224236344448604, 3.43551399195511712236903657816, 5.12553753782728504599967213955, 5.64713656488976520025843190553, 7.10655360964114683232862426342, 7.25821217549352394679105745722, 7.37732792484842868284117396695, 8.435416633558901220546111564259, 9.454139797789076397700060710186, 9.946388780896094143993725974526, 9.998802688611086109464520430262, 10.14284762309064775210492991809, 10.99748335226301824877763326540, 11.90327030397803109767832616100, 12.44501284768504795812830520917, 12.52412358353080258310017563007, 13.11981359757681676277581925418, 13.17891789008517935944615106046, 14.13425557388889850963185316499, 14.70834586642623482984680703603, 14.77321029738825350425838697111, 15.15826741594825591264719986097, 15.94156673157555611824397647322, 16.03801878611397430936717703502, 16.47192746263695183583764465299

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