Properties

Label 4-450e2-1.1-c5e2-0-5
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·4-s − 384·11-s + 256·16-s + 5.48e3·19-s + 1.18e4·29-s − 1.37e4·31-s + 756·41-s + 6.14e3·44-s + 1.96e4·49-s − 6.99e4·59-s − 1.96e4·61-s − 4.09e3·64-s − 1.40e5·71-s − 8.76e4·76-s − 9.04e3·79-s + 7.69e4·89-s − 1.55e5·101-s − 4.13e5·109-s − 1.89e5·116-s − 2.11e5·121-s + 2.19e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.956·11-s + 1/4·16-s + 3.48·19-s + 2.60·29-s − 2.56·31-s + 0.0702·41-s + 0.478·44-s + 1.17·49-s − 2.61·59-s − 0.677·61-s − 1/8·64-s − 3.30·71-s − 1.74·76-s − 0.162·79-s + 1.03·89-s − 1.51·101-s − 3.33·109-s − 1.30·116-s − 1.31·121-s + 1.28·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-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: induced by $\chi_{450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.428823547\)
\(L(\frac12)\) \(\approx\) \(1.428823547\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{4} T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 19690 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 192 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 480650 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2259070 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 2740 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10420330 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 5910 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6868 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 108239590 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 378 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288092530 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 286503130 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 752228710 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 34980 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 9838 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1563076930 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 70212 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3662758990 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 4520 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4019056190 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 38490 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 17171001790 T^{2} + p^{10} T^{4} \)
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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

−10.76656733887670664673149642774, −10.08241192306380424153451880554, −9.557455293700342535705976263121, −9.061787114554650116522998804702, −9.004276915383647133073234122526, −8.065596996378590770652163928973, −7.81100620797478577951178630402, −7.33386643894994979579433650161, −7.05922621110082968527470876357, −6.17541032039696109384632373287, −5.67673421656492059811411036886, −5.24077279646477479109744971849, −4.92878701657058044852398638875, −4.26548087935373380214980896378, −3.55652948421125300300334901107, −2.86157357112046419438571410788, −2.83459954782321005906137828440, −1.48842805208648658581262739744, −1.18624832241486541605750409412, −0.31281360253494608468916559672, 0.31281360253494608468916559672, 1.18624832241486541605750409412, 1.48842805208648658581262739744, 2.83459954782321005906137828440, 2.86157357112046419438571410788, 3.55652948421125300300334901107, 4.26548087935373380214980896378, 4.92878701657058044852398638875, 5.24077279646477479109744971849, 5.67673421656492059811411036886, 6.17541032039696109384632373287, 7.05922621110082968527470876357, 7.33386643894994979579433650161, 7.81100620797478577951178630402, 8.065596996378590770652163928973, 9.004276915383647133073234122526, 9.061787114554650116522998804702, 9.557455293700342535705976263121, 10.08241192306380424153451880554, 10.76656733887670664673149642774

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