Properties

Label 4-450e2-1.1-c5e2-0-20
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $5208.90$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 48·4-s + 114·7-s − 256·8-s − 174·11-s + 642·13-s − 912·14-s + 1.28e3·16-s − 1.51e3·17-s − 120·19-s + 1.39e3·22-s − 4.35e3·23-s − 5.13e3·26-s + 5.47e3·28-s + 2.43e3·29-s + 1.16e4·31-s − 6.14e3·32-s + 1.21e4·34-s − 1.75e4·37-s + 960·38-s − 2.49e4·41-s − 2.41e4·43-s − 8.35e3·44-s + 3.48e4·46-s + 1.33e4·47-s + 7.35e3·49-s + 3.08e4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.879·7-s − 1.41·8-s − 0.433·11-s + 1.05·13-s − 1.24·14-s + 5/4·16-s − 1.27·17-s − 0.0762·19-s + 0.613·22-s − 1.71·23-s − 1.49·26-s + 1.31·28-s + 0.536·29-s + 2.18·31-s − 1.06·32-s + 1.79·34-s − 2.11·37-s + 0.107·38-s − 2.32·41-s − 1.99·43-s − 0.650·44-s + 2.42·46-s + 0.879·47-s + 0.437·49-s + 1.58·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5208.90\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 202500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{2} T )^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 114 T + 5638 T^{2} - 114 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 174 T + 298446 T^{2} + 174 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 642 T + 564602 T^{2} - 642 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1514 T + 3131738 T^{2} + 1514 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 120 T + 459398 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 4352 T + 13111262 T^{2} + 4352 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 2430 T + 19735498 T^{2} - 2430 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 11684 T + 90263166 T^{2} - 11684 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 17586 T + 204732538 T^{2} + 17586 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 24984 T + 351666366 T^{2} + 24984 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 24168 T + 342118342 T^{2} + 24168 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 13316 T + 501894878 T^{2} - 13316 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 13698 T + 498576562 T^{2} - 13698 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 23730 T + 501951198 T^{2} - 23730 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 57124 T + 2217210846 T^{2} - 57124 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 38316 T + 2836339078 T^{2} + 38316 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 156 p T + 940164046 T^{2} - 156 p^{6} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 88548 T + 5806445362 T^{2} + 88548 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 180 p T + 4980519998 T^{2} - 180 p^{6} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 50792 T + 2695703702 T^{2} + 50792 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 60420 T + 10831762998 T^{2} - 60420 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 171216 T + 24495416578 T^{2} + 171216 p^{5} T^{3} + p^{10} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.961897107684374835223184260867, −9.911755383800277343776432842060, −8.848290827137713978174912253748, −8.598986492867825712336086994691, −8.269197551004626213009340533446, −8.256385384360324322744808016710, −7.30147171637064655759119770898, −6.98831381067407799977136733548, −6.35351662166181708945168001601, −6.19076695435597970548512202407, −5.19401456646129692521167591079, −5.00575888515818835319304954151, −4.03904863230108983270368989017, −3.65637276813871532229859876817, −2.69337983956829755186121617241, −2.25009107965815245076399840781, −1.51924495995894136039145930498, −1.22169472352998555175481954241, 0, 0, 1.22169472352998555175481954241, 1.51924495995894136039145930498, 2.25009107965815245076399840781, 2.69337983956829755186121617241, 3.65637276813871532229859876817, 4.03904863230108983270368989017, 5.00575888515818835319304954151, 5.19401456646129692521167591079, 6.19076695435597970548512202407, 6.35351662166181708945168001601, 6.98831381067407799977136733548, 7.30147171637064655759119770898, 8.256385384360324322744808016710, 8.269197551004626213009340533446, 8.598986492867825712336086994691, 8.848290827137713978174912253748, 9.911755383800277343776432842060, 9.961897107684374835223184260867

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