Properties

Label 4-45e2-1.1-c7e2-0-0
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $197.608$
Root an. cond. $3.74931$
Motivic weight $7$
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  + 251·4-s + 4.66e4·16-s − 2.31e4·19-s − 7.81e4·25-s + 4.13e5·31-s + 1.64e6·49-s − 5.54e6·61-s + 7.58e6·64-s − 5.82e6·76-s + 1.75e7·79-s − 1.96e7·100-s + 4.91e7·109-s − 3.89e7·121-s + 1.03e8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.25e8·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.96·4-s + 2.84·16-s − 0.775·19-s − 25-s + 2.49·31-s + 2·49-s − 3.13·61-s + 3.61·64-s − 1.52·76-s + 3.99·79-s − 1.96·100-s + 3.63·109-s − 2·121-s + 4.88·124-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(197.608\)
Root analytic conductor: \(3.74931\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(4.423464662\)
\(L(\frac12)\) \(\approx\) \(4.423464662\)
\(L(\frac{9}{2})\) 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$C_2$ \( 1 + p^{7} T^{2} \)
good2$C_2^2$ \( 1 - 251 T^{2} + p^{14} T^{4} \)
7$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 139543474 T^{2} + p^{14} T^{4} \)
19$C_2$ \( ( 1 + 11596 T + p^{7} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3027622786 T^{2} + p^{14} T^{4} \)
29$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 206648 T + p^{7} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 877116235954 T^{2} + p^{14} T^{4} \)
53$C_2^2$ \( 1 + 787198687546 T^{2} + p^{14} T^{4} \)
59$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2774518 T + p^{7} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8763536 T + p^{7} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 48542379188534 T^{2} + p^{14} T^{4} \)
89$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
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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

−14.96682209715894523539042530149, −13.96652610283909232195547179747, −13.61213728035192569625942589724, −12.59672911506627904153534246594, −12.05620169315300482689843263320, −11.81575929398353467876073530987, −11.01529541255839823890584698733, −10.52440268281559011093358239522, −10.08517336570608779436937350574, −9.145955043120480033933408489319, −8.116915225337276926602427628914, −7.73076961211392014063059499755, −6.91966800162401708049228780662, −6.26964472596655579337061068244, −5.86616801146995672539025550945, −4.63201554311879114118881029484, −3.49187975489863442849944411752, −2.59527483843023257786833472381, −1.94357710108871862428861832509, −0.870243483062542172968589152767, 0.870243483062542172968589152767, 1.94357710108871862428861832509, 2.59527483843023257786833472381, 3.49187975489863442849944411752, 4.63201554311879114118881029484, 5.86616801146995672539025550945, 6.26964472596655579337061068244, 6.91966800162401708049228780662, 7.73076961211392014063059499755, 8.116915225337276926602427628914, 9.145955043120480033933408489319, 10.08517336570608779436937350574, 10.52440268281559011093358239522, 11.01529541255839823890584698733, 11.81575929398353467876073530987, 12.05620169315300482689843263320, 12.59672911506627904153534246594, 13.61213728035192569625942589724, 13.96652610283909232195547179747, 14.96682209715894523539042530149

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