Properties

Label 4-450e2-1.1-c2e2-0-3
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $150.347$
Root an. cond. $3.50165$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 16·7-s + 8·11-s + 6·13-s − 32·14-s − 4·16-s − 38·17-s + 16·22-s + 40·23-s + 12·26-s − 32·28-s − 88·31-s − 8·32-s − 76·34-s + 6·37-s + 140·41-s − 72·43-s + 16·44-s + 80·46-s + 128·49-s + 12·52-s − 34·53-s + 144·61-s − 176·62-s − 8·64-s − 88·67-s + ⋯
L(s)  = 1  + 2-s + 1/2·4-s − 2.28·7-s + 8/11·11-s + 6/13·13-s − 2.28·14-s − 1/4·16-s − 2.23·17-s + 8/11·22-s + 1.73·23-s + 6/13·26-s − 8/7·28-s − 2.83·31-s − 1/4·32-s − 2.23·34-s + 6/37·37-s + 3.41·41-s − 1.67·43-s + 4/11·44-s + 1.73·46-s + 2.61·49-s + 3/13·52-s − 0.641·53-s + 2.36·61-s − 2.83·62-s − 1/8·64-s − 1.31·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(150.347\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.685882721\)
\(L(\frac12)\) \(\approx\) \(1.685882721\)
\(L(2)\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 + 38 T + 722 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 658 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 40 T + 800 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 238 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 72 T + 2592 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 + 1502 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 72 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 88 T + 3872 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 + 88 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 110 T + 6050 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 12338 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 15166 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 114 T + 6498 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \)
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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

−11.16735063093424290702195738466, −10.79785339661914097149225309578, −10.29698093800720513732218259981, −9.665430619152321039611415685836, −9.120781526493720330849209139236, −8.951584402564481257549442833166, −8.858544468615040273682033763957, −7.61179383011849509141596673671, −7.12654776820321573802488581568, −6.85815010741146095588493947741, −6.20086433134799526670751983998, −6.17550557877759553897974421321, −5.45469971713568904307878173902, −4.78053673542332195963079660501, −4.01692006462026777210756038754, −3.88639314077383374462759074766, −3.06481587553087848869698363139, −2.75910930754673675747905998697, −1.78827824898256082582368843122, −0.44305221038273263766191932166, 0.44305221038273263766191932166, 1.78827824898256082582368843122, 2.75910930754673675747905998697, 3.06481587553087848869698363139, 3.88639314077383374462759074766, 4.01692006462026777210756038754, 4.78053673542332195963079660501, 5.45469971713568904307878173902, 6.17550557877759553897974421321, 6.20086433134799526670751983998, 6.85815010741146095588493947741, 7.12654776820321573802488581568, 7.61179383011849509141596673671, 8.858544468615040273682033763957, 8.951584402564481257549442833166, 9.120781526493720330849209139236, 9.665430619152321039611415685836, 10.29698093800720513732218259981, 10.79785339661914097149225309578, 11.16735063093424290702195738466

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