Properties

Label 4-45e2-1.1-c5e2-0-2
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $52.0890$
Root an. cond. $2.68649$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 20·4-s + 90·5-s − 504·11-s − 624·16-s − 440·19-s + 1.80e3·20-s + 4.97e3·25-s + 1.38e4·29-s + 1.35e4·31-s + 396·41-s − 1.00e4·44-s + 3.00e4·49-s − 4.53e4·55-s + 4.93e4·59-s − 1.13e4·61-s − 3.29e4·64-s − 1.06e5·71-s − 8.80e3·76-s + 1.03e5·79-s − 5.61e4·80-s + 1.99e4·89-s − 3.96e4·95-s + 9.95e4·100-s + 2.18e5·101-s − 4.20e4·109-s + 2.77e5·116-s − 1.31e5·121-s + ⋯
L(s)  = 1  + 5/8·4-s + 1.60·5-s − 1.25·11-s − 0.609·16-s − 0.279·19-s + 1.00·20-s + 1.59·25-s + 3.06·29-s + 2.52·31-s + 0.0367·41-s − 0.784·44-s + 1.78·49-s − 2.02·55-s + 1.84·59-s − 0.392·61-s − 1.00·64-s − 2.51·71-s − 0.174·76-s + 1.87·79-s − 0.981·80-s + 0.267·89-s − 0.450·95-s + 0.994·100-s + 2.12·101-s − 0.338·109-s + 1.91·116-s − 0.817·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(52.0890\)
Root analytic conductor: \(2.68649\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.263728206\)
\(L(\frac12)\) \(\approx\) \(3.263728206\)
\(L(\frac{7}{2})\) 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)$
bad3 \( 1 \)
5$C_2$ \( 1 - 18 p T + p^{5} T^{2} \)
good2$C_2^2$ \( 1 - 5 p^{2} T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 30050 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 252 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 728330 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2363810 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6946370 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 6930 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6752 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56462470 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 198 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 293842250 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 347593490 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 802472090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 24660 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 795787610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 53352 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 883886830 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 51920 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4053674810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 9990 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6923133890 T^{2} + p^{10} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.37328280401321787132936946656, −14.32495061373711937131605361349, −13.88533960379769354094494703085, −13.44870478713871244123565871646, −12.98414098550329076800317843724, −12.07099147048482634046765730285, −11.73104783609597120170388968097, −10.57114513803079156106554651776, −10.34247707015070465266509454029, −9.964282482124206858491416597958, −8.911885214124723887739297804694, −8.418048433651762159224634439680, −7.48808201075511211134016280125, −6.44244651590754007683773395272, −6.31654591453785310420190729381, −5.22676072222744257576886987393, −4.57444132457676853726832937226, −2.64897320938473199539532211455, −2.49090330731543460257831706405, −1.02087537862008371301362993346, 1.02087537862008371301362993346, 2.49090330731543460257831706405, 2.64897320938473199539532211455, 4.57444132457676853726832937226, 5.22676072222744257576886987393, 6.31654591453785310420190729381, 6.44244651590754007683773395272, 7.48808201075511211134016280125, 8.418048433651762159224634439680, 8.911885214124723887739297804694, 9.964282482124206858491416597958, 10.34247707015070465266509454029, 10.57114513803079156106554651776, 11.73104783609597120170388968097, 12.07099147048482634046765730285, 12.98414098550329076800317843724, 13.44870478713871244123565871646, 13.88533960379769354094494703085, 14.32495061373711937131605361349, 15.37328280401321787132936946656

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