Properties

Label 4-45e2-1.1-c3e2-0-0
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $7.04948$
Root an. cond. $1.62944$
Motivic weight $3$
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  + 11·4-s + 57·16-s + 328·19-s − 125·25-s − 464·31-s + 686·49-s − 716·61-s − 77·64-s + 3.60e3·76-s − 608·79-s − 1.37e3·100-s − 3.66e3·109-s − 2.66e3·121-s − 5.10e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 11/8·4-s + 0.890·16-s + 3.96·19-s − 25-s − 2.68·31-s + 2·49-s − 1.50·61-s − 0.150·64-s + 5.44·76-s − 0.865·79-s − 1.37·100-s − 3.22·109-s − 2·121-s − 3.69·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(7.04948\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.212987015\)
\(L(\frac12)\) \(\approx\) \(2.212987015\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{3} T^{2} \)
good2$C_2^2$ \( 1 - 11 T^{2} + p^{6} T^{4} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 9394 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 14654 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 232 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 90034 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 88666 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 358 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 304 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 469546 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
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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

−15.74452160286446100211032089098, −15.22154044912984405789724003881, −14.53397201448028164417709467814, −13.84699594673982514302232874476, −13.51374791651967017059918599848, −12.51570167502749234044992463822, −11.87494613724665095041654224346, −11.63856618079614887931866362390, −10.99865052595189472039068516768, −10.34108650582523083908659026227, −9.495787169864354627196298885071, −9.134228034675816325519154653218, −7.64268166337462406338082967422, −7.55886491435439315799155592198, −6.87583696584929135726137543029, −5.68419233400361906279815777704, −5.39726167887966620649887535723, −3.71445653822306625951668107148, −2.84799502217423045744567467775, −1.48944059142701723386272575823, 1.48944059142701723386272575823, 2.84799502217423045744567467775, 3.71445653822306625951668107148, 5.39726167887966620649887535723, 5.68419233400361906279815777704, 6.87583696584929135726137543029, 7.55886491435439315799155592198, 7.64268166337462406338082967422, 9.134228034675816325519154653218, 9.495787169864354627196298885071, 10.34108650582523083908659026227, 10.99865052595189472039068516768, 11.63856618079614887931866362390, 11.87494613724665095041654224346, 12.51570167502749234044992463822, 13.51374791651967017059918599848, 13.84699594673982514302232874476, 14.53397201448028164417709467814, 15.22154044912984405789724003881, 15.74452160286446100211032089098

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