Properties

Degree 4
Conductor $ 5^{2} $
Sign $1$
Motivic weight 7
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·2-s + 20·3-s + 120·4-s − 250·5-s + 400·6-s − 100·7-s − 640·8-s + 790·9-s − 5.00e3·10-s + 4.54e3·11-s + 2.40e3·12-s + 3.54e3·13-s − 2.00e3·14-s − 5.00e3·15-s − 1.15e4·16-s − 2.73e4·17-s + 1.58e4·18-s + 3.87e4·19-s − 3.00e4·20-s − 2.00e3·21-s + 9.08e4·22-s − 1.24e5·23-s − 1.28e4·24-s + 4.68e4·25-s + 7.08e4·26-s + 6.73e4·27-s − 1.20e4·28-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.427·3-s + 0.937·4-s − 0.894·5-s + 0.756·6-s − 0.110·7-s − 0.441·8-s + 0.361·9-s − 1.58·10-s + 1.02·11-s + 0.400·12-s + 0.446·13-s − 0.194·14-s − 0.382·15-s − 0.707·16-s − 1.34·17-s + 0.638·18-s + 1.29·19-s − 0.838·20-s − 0.0471·21-s + 1.81·22-s − 2.12·23-s − 0.189·24-s + 3/5·25-s + 0.789·26-s + 0.658·27-s − 0.103·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 25,\ (\ :7/2, 7/2),\ 1)$
$L(4)$  $\approx$  $2.61347$
$L(\frac12)$  $\approx$  $2.61347$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 4. If $p = 5$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$ \( ( 1 + p^{3} T )^{2} \)
good2$D_{4}$ \( 1 - 5 p^{2} T + 35 p^{3} T^{2} - 5 p^{9} T^{3} + p^{14} T^{4} \)
3$D_{4}$ \( 1 - 20 T - 130 p T^{2} - 20 p^{7} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 + 100 T + 1411250 T^{2} + 100 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 - 4544 T + 31976326 T^{2} - 4544 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 - 3540 T + 100535470 T^{2} - 3540 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 + 27340 T + 901005190 T^{2} + 27340 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 - 2040 p T + 113449762 p T^{2} - 2040 p^{8} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 124140 T + 10649684530 T^{2} + 124140 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 72260 T + 6846819118 T^{2} + 72260 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 306824 T + 77964629966 T^{2} - 306824 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 123020 T + 144088599870 T^{2} + 123020 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 264364 T + 161786388886 T^{2} - 264364 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 423300 T + 446651231050 T^{2} - 423300 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 + 105460 T + 858715356610 T^{2} + 105460 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 2391580 T + 3562552504510 T^{2} + 2391580 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 + 1120120 T + 1362334883638 T^{2} + 1120120 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 - 2257044 T + 5613447576526 T^{2} - 2257044 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 4516460 T + 16742087664890 T^{2} - 4516460 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 621784 T + 17914494152446 T^{2} - 621784 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 - 4569060 T + 23424949855030 T^{2} - 4569060 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 4333040 T + 330830231042 p T^{2} - 4333040 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 + 9793020 T + 59971104320890 T^{2} + 9793020 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 6025620 T + 89865866149558 T^{2} - 6025620 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 4609540 T + 142930351581510 T^{2} - 4609540 p^{7} T^{3} + p^{14} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−22.60501002348992296283196614035, −22.54060403502313847767575986943, −21.90639216795998310222946270991, −20.88966920745470466595519261826, −20.05309366556374309499376322086, −19.52897156300637550527526803721, −18.41503997148447739979704080654, −17.49591101229876690110058287826, −15.85325839574730429144282105618, −15.70441842117171651071931414555, −14.21000976480262357162330942802, −14.15821035961646171329220384062, −13.04321206354133836289379117522, −12.19035669906091329910107758905, −11.34895122175493616646545967309, −9.539264410826295415867260441329, −8.132385337579659451415013951776, −6.41028214690635329556348984161, −4.56624038341865425167686208504, −3.63715009395516665553552623709, 3.63715009395516665553552623709, 4.56624038341865425167686208504, 6.41028214690635329556348984161, 8.132385337579659451415013951776, 9.539264410826295415867260441329, 11.34895122175493616646545967309, 12.19035669906091329910107758905, 13.04321206354133836289379117522, 14.15821035961646171329220384062, 14.21000976480262357162330942802, 15.70441842117171651071931414555, 15.85325839574730429144282105618, 17.49591101229876690110058287826, 18.41503997148447739979704080654, 19.52897156300637550527526803721, 20.05309366556374309499376322086, 20.88966920745470466595519261826, 21.90639216795998310222946270991, 22.54060403502313847767575986943, 22.60501002348992296283196614035

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