Properties

Label 4-45e2-1.1-c4e2-0-1
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $21.6378$
Root an. cond. $2.15676$
Motivic weight $4$
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  − 10·2-s + 50·4-s + 50·5-s + 80·7-s − 160·8-s − 500·10-s − 200·11-s + 410·13-s − 800·14-s + 444·16-s + 470·17-s + 2.50e3·20-s + 2.00e3·22-s + 680·23-s + 1.87e3·25-s − 4.10e3·26-s + 4.00e3·28-s + 856·31-s − 1.88e3·32-s − 4.70e3·34-s + 4.00e3·35-s − 1.51e3·37-s − 8.00e3·40-s + 1.90e3·41-s − 2.44e3·43-s − 1.00e4·44-s − 6.80e3·46-s + ⋯
L(s)  = 1  − 5/2·2-s + 25/8·4-s + 2·5-s + 1.63·7-s − 5/2·8-s − 5·10-s − 1.65·11-s + 2.42·13-s − 4.08·14-s + 1.73·16-s + 1.62·17-s + 25/4·20-s + 4.13·22-s + 1.28·23-s + 3·25-s − 6.06·26-s + 5.10·28-s + 0.890·31-s − 1.83·32-s − 4.06·34-s + 3.26·35-s − 1.10·37-s − 5·40-s + 1.13·41-s − 1.31·43-s − 5.16·44-s − 3.21·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(21.6378\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.025863956\)
\(L(\frac12)\) \(\approx\) \(1.025863956\)
\(L(3)\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - p^{2} T )^{2} \)
good2$C_2^2$ \( 1 + 5 p T + 25 p T^{2} + 5 p^{5} T^{3} + p^{8} T^{4} \)
7$C_2^2$ \( 1 - 80 T + 3200 T^{2} - 80 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2$ \( ( 1 + 100 T + p^{4} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 410 T + 84050 T^{2} - 410 p^{4} T^{3} + p^{8} T^{4} \)
17$C_2^2$ \( 1 - 470 T + 110450 T^{2} - 470 p^{4} T^{3} + p^{8} T^{4} \)
19$C_2^2$ \( 1 - 255458 T^{2} + p^{8} T^{4} \)
23$C_2^2$ \( 1 - 680 T + 231200 T^{2} - 680 p^{4} T^{3} + p^{8} T^{4} \)
29$C_2^2$ \( 1 - 1212062 T^{2} + p^{8} T^{4} \)
31$C_2$ \( ( 1 - 428 T + p^{4} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 1510 T + 1140050 T^{2} + 1510 p^{4} T^{3} + p^{8} T^{4} \)
41$C_2$ \( ( 1 - 950 T + p^{4} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 2440 T + 2976800 T^{2} + 2440 p^{4} T^{3} + p^{8} T^{4} \)
47$C_2^2$ \( 1 + 640 T + 204800 T^{2} + 640 p^{4} T^{3} + p^{8} T^{4} \)
53$C_2^2$ \( 1 - 1010 T + 510050 T^{2} - 1010 p^{4} T^{3} + p^{8} T^{4} \)
59$C_2^2$ \( 1 + 15455278 T^{2} + p^{8} T^{4} \)
61$C_2$ \( ( 1 + 3808 T + p^{4} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 680 T + 231200 T^{2} - 680 p^{4} T^{3} + p^{8} T^{4} \)
71$C_2$ \( ( 1 + 3400 T + p^{4} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 830 T + 344450 T^{2} - 830 p^{4} T^{3} + p^{8} T^{4} \)
79$C_2^2$ \( 1 - 32580338 T^{2} + p^{8} T^{4} \)
83$C_2^2$ \( 1 + 1360 T + 924800 T^{2} + 1360 p^{4} T^{3} + p^{8} T^{4} \)
89$C_2^2$ \( 1 - 120421982 T^{2} + p^{8} T^{4} \)
97$C_2^2$ \( 1 - 3230 T + 5216450 T^{2} - 3230 p^{4} T^{3} + p^{8} T^{4} \)
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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

−15.60902321995802157718247358034, −14.93335667345749484841391268839, −14.22887761342581863350793123036, −13.60469477727313725774403377547, −13.23243016215544772796885912275, −12.32798336888523913000149918694, −11.13863414495670208643152394214, −10.87071480473455952488790464967, −10.27816425535131573361315562552, −10.05764281745735347802030116292, −8.978313391807621173776114886713, −8.842516389332419937877742541108, −8.063258574828759974350640315673, −7.74570646486791488147425204999, −6.56798480812039893955511375980, −5.61584908961168104185922907072, −5.18008670430117112402853184422, −2.88303186380730606771898343364, −1.39363567087694261239185012619, −1.29531835230208170270457325946, 1.29531835230208170270457325946, 1.39363567087694261239185012619, 2.88303186380730606771898343364, 5.18008670430117112402853184422, 5.61584908961168104185922907072, 6.56798480812039893955511375980, 7.74570646486791488147425204999, 8.063258574828759974350640315673, 8.842516389332419937877742541108, 8.978313391807621173776114886713, 10.05764281745735347802030116292, 10.27816425535131573361315562552, 10.87071480473455952488790464967, 11.13863414495670208643152394214, 12.32798336888523913000149918694, 13.23243016215544772796885912275, 13.60469477727313725774403377547, 14.22887761342581863350793123036, 14.93335667345749484841391268839, 15.60902321995802157718247358034

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