Properties

Label 4-450e2-1.1-c5e2-0-13
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 − 444·11-s + 256·16-s − 3.37e3·19-s + 1.57e4·29-s − 1.71e4·31-s + 3.63e4·41-s + 7.10e3·44-s + 3.14e4·49-s + 3.52e4·59-s − 4.37e4·61-s − 4.09e3·64-s + 8.14e4·71-s + 5.39e4·76-s + 1.38e5·79-s + 7.00e4·89-s + 6.76e4·101-s + 4.34e5·109-s − 2.52e5·116-s − 1.74e5·121-s + 2.74e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.10·11-s + 1/4·16-s − 2.14·19-s + 3.48·29-s − 3.21·31-s + 3.37·41-s + 0.553·44-s + 1.86·49-s + 1.31·59-s − 1.50·61-s − 1/8·64-s + 1.91·71-s + 1.07·76-s + 2.49·79-s + 0.937·89-s + 0.659·101-s + 3.49·109-s − 1.74·116-s − 1.08·121-s + 1.60·124-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.497404398\)
\(L(\frac12)\) \(\approx\) \(2.497404398\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 - 31405 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 222 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 732385 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2813470 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 1685 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 12779050 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 7890 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8593 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 64003750 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 18168 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 88065685 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 457484410 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 515407930 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 17610 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 21853 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2700238765 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 40728 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2941636750 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 69160 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 3901547870 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 35040 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 17174003185 T^{2} + p^{10} T^{4} \)
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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

−10.43524768640312594995690423099, −10.36572198410571433259244319562, −9.409752495331592442034511686926, −9.255362625122993155293963565153, −8.519512774047546782163230570283, −8.499062591484156708715608330817, −7.66848001324565883663412994068, −7.52763011083938784428574055157, −6.75867316304925232062080097300, −6.25783472320988400852693215564, −5.84689711890490573158909254234, −5.21927576015611910427867213164, −4.76168897673030163512005715634, −4.20313586494754919300679895789, −3.81616559935190045537131504160, −2.96536391751812507813815874103, −2.33773059429520049498204790659, −2.01282817183255031494802720666, −0.71150664222530705688436511650, −0.57378078154384954054164338439, 0.57378078154384954054164338439, 0.71150664222530705688436511650, 2.01282817183255031494802720666, 2.33773059429520049498204790659, 2.96536391751812507813815874103, 3.81616559935190045537131504160, 4.20313586494754919300679895789, 4.76168897673030163512005715634, 5.21927576015611910427867213164, 5.84689711890490573158909254234, 6.25783472320988400852693215564, 6.75867316304925232062080097300, 7.52763011083938784428574055157, 7.66848001324565883663412994068, 8.499062591484156708715608330817, 8.519512774047546782163230570283, 9.255362625122993155293963565153, 9.409752495331592442034511686926, 10.36572198410571433259244319562, 10.43524768640312594995690423099

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