Properties

Label 4-45e2-1.1-c13e2-0-0
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $2328.44$
Root an. cond. $6.94650$
Motivic weight $13$
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  + 2.33e3·4-s − 6.16e7·16-s + 7.07e7·19-s − 1.22e9·25-s − 1.65e10·31-s + 1.93e11·49-s + 1.60e12·61-s − 3.01e11·64-s + 1.65e11·76-s − 7.88e12·79-s − 2.85e12·100-s − 8.96e12·109-s − 6.90e13·121-s − 3.87e13·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.05e14·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 0.285·4-s − 0.918·16-s + 0.344·19-s − 25-s − 3.35·31-s + 2·49-s + 3.98·61-s − 0.547·64-s + 0.0984·76-s − 3.64·79-s − 0.285·100-s − 0.511·109-s − 2·121-s − 0.958·124-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2328.44\)
Root analytic conductor: \(6.94650\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.086747252\)
\(L(\frac12)\) \(\approx\) \(1.086747252\)
\(L(\frac{15}{2})\) 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$C_2$ \( 1 + p^{13} T^{2} \)
good2$C_2^2$ \( 1 - 2339 T^{2} + p^{26} T^{4} \)
7$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 13734556739205106 T^{2} + p^{26} T^{4} \)
19$C_2$ \( ( 1 - 35371436 T + p^{13} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 991340034832255646 T^{2} + p^{26} T^{4} \)
29$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8289908632 T + p^{13} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
47$C_2^2$ \( 1 - \)\(10\!\cdots\!34\)\( T^{2} + p^{26} T^{4} \)
53$C_2^2$ \( 1 + \)\(13\!\cdots\!34\)\( T^{2} + p^{26} T^{4} \)
59$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 800891666642 T + p^{13} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 3942239820496 T + p^{13} T^{2} )^{2} \)
83$C_2^2$ \( 1 + \)\(45\!\cdots\!54\)\( T^{2} + p^{26} T^{4} \)
89$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
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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

−14.41464381791894689139629181909, −13.04432622927379629281613699531, −12.03275485218143254812009839263, −11.39298891988600496482615306685, −11.16267326194968680767251463718, −10.29662524680622907463345713528, −9.763783047402253697368486827026, −8.968856242876960147835276474051, −8.650832246927276787837221334047, −7.52392521483161377855136521549, −7.27228843444565698245690701711, −6.53358358164771606295739295399, −5.55634701639146730000838627436, −5.31960770612100537467410957328, −4.04548652805962883467135612711, −3.77967192565804348670268384470, −2.63271769106929173238347451132, −2.06847112521517763565442339962, −1.31522854816042370574673025240, −0.27814166767114872991654230387, 0.27814166767114872991654230387, 1.31522854816042370574673025240, 2.06847112521517763565442339962, 2.63271769106929173238347451132, 3.77967192565804348670268384470, 4.04548652805962883467135612711, 5.31960770612100537467410957328, 5.55634701639146730000838627436, 6.53358358164771606295739295399, 7.27228843444565698245690701711, 7.52392521483161377855136521549, 8.650832246927276787837221334047, 8.968856242876960147835276474051, 9.763783047402253697368486827026, 10.29662524680622907463345713528, 11.16267326194968680767251463718, 11.39298891988600496482615306685, 12.03275485218143254812009839263, 13.04432622927379629281613699531, 14.41464381791894689139629181909

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