Properties

Label 4-45e2-1.1-c5e2-0-0
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 61·4-s + 2.69e3·16-s − 4.32e3·19-s − 3.12e3·25-s − 1.63e4·31-s + 3.36e4·49-s + 6.96e4·61-s − 1.02e5·64-s + 2.64e5·76-s + 1.40e5·79-s + 1.90e5·100-s − 2.51e5·109-s − 3.22e5·121-s + 9.94e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.42e5·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.90·4-s + 2.63·16-s − 2.75·19-s − 25-s − 3.04·31-s + 2·49-s + 2.39·61-s − 3.11·64-s + 5.24·76-s + 2.52·79-s + 1.90·100-s − 2.03·109-s − 2·121-s + 5.80·124-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 + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-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

\(L(3)\) \(\approx\) \(0.5214581548\)
\(L(\frac12)\) \(\approx\) \(0.5214581548\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{5} T^{2} \)
good2$C_2^2$ \( 1 + 61 T^{2} + p^{10} T^{4} \)
7$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2419214 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 2164 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10950686 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8152 T + p^{5} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 311808014 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 + 836229514 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 34802 T + p^{5} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 70064 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2642233286 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
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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

−15.47073149299784855289323554288, −14.82572066932156373207359943628, −13.92000085222676721814146376554, −13.31928974857317336030878198203, −12.76328507170351184352843388663, −12.63585252534962458022754457820, −11.63550193010177143657278327590, −10.55751822592347745645408452783, −10.47894951463602710318976746051, −9.282464103848830981676106688234, −9.167404447295462908416571646246, −8.386053351569183103621742097455, −7.86360750230412645713720487327, −6.79694444783445960510253927693, −5.74707649349879507936927578801, −5.18568470146185760372901820855, −3.98593963029854148127895255461, −3.94012057567258160197994312235, −2.04594132536270335338382510731, −0.39239110803778337893239544639, 0.39239110803778337893239544639, 2.04594132536270335338382510731, 3.94012057567258160197994312235, 3.98593963029854148127895255461, 5.18568470146185760372901820855, 5.74707649349879507936927578801, 6.79694444783445960510253927693, 7.86360750230412645713720487327, 8.386053351569183103621742097455, 9.167404447295462908416571646246, 9.282464103848830981676106688234, 10.47894951463602710318976746051, 10.55751822592347745645408452783, 11.63550193010177143657278327590, 12.63585252534962458022754457820, 12.76328507170351184352843388663, 13.31928974857317336030878198203, 13.92000085222676721814146376554, 14.82572066932156373207359943628, 15.47073149299784855289323554288

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