Properties

Label 4-75e2-1.1-c8e2-0-2
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $933.509$
Root an. cond. $5.52751$
Motivic weight $8$
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  + 223·4-s − 6.56e3·9-s − 1.58e4·16-s + 4.07e5·19-s + 3.66e6·31-s − 1.46e6·36-s − 1.15e7·49-s + 2.86e7·61-s − 1.81e7·64-s + 9.09e7·76-s + 1.39e8·79-s + 4.30e7·81-s − 5.19e8·109-s + 4.28e8·121-s + 8.16e8·124-s + 127-s + 131-s + 137-s + 139-s + 1.03e8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.63e9·169-s − 2.67e9·171-s + ⋯
L(s)  = 1  + 0.871·4-s − 9-s − 0.241·16-s + 3.13·19-s + 3.96·31-s − 0.871·36-s − 2·49-s + 2.06·61-s − 1.08·64-s + 2.72·76-s + 3.57·79-s + 81-s − 3.67·109-s + 2·121-s + 3.45·124-s + 0.241·144-s − 2·169-s − 3.13·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(933.509\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.860452937\)
\(L(\frac12)\) \(\approx\) \(3.860452937\)
\(L(5)\) 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$C_2$ \( 1 + p^{8} T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 - 223 T^{2} + p^{16} T^{4} \)
7$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
13$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 13505544958 T^{2} + p^{16} T^{4} \)
19$C_2$ \( ( 1 - 203998 T + p^{8} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 145963835522 T^{2} + p^{16} T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
31$C_2$ \( ( 1 - 1831682 T + p^{8} T^{2} )^{2} \)
37$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
43$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 17436524386562 T^{2} + p^{16} T^{4} \)
53$C_2^2$ \( 1 + 34736892000962 T^{2} + p^{16} T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
61$C_2$ \( ( 1 - 14324642 T + p^{8} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
73$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 69617278 T + p^{8} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4489783501049278 T^{2} + p^{16} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
97$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
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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

−13.54420653410565103323398183751, −12.20709965402453296629993409195, −12.03463020386680539693085394798, −11.42103550288830668724248085958, −11.28625091630429291855988155624, −10.36121985185948969219155767194, −9.712297620810837928519282221089, −9.425374975517169057412823621140, −8.374432549960790948234631982260, −8.020341811852842834617252662091, −7.35988039820324632836249891454, −6.54040448354097979665897131091, −6.24385332469594188633991352750, −5.22683797328678044307727658321, −4.88738845624000181590951079640, −3.60242279634992969322859654286, −2.88730279620532147633317227312, −2.49637904898507429020512654708, −1.26365004012369579093150663173, −0.66897873447030978731638254418, 0.66897873447030978731638254418, 1.26365004012369579093150663173, 2.49637904898507429020512654708, 2.88730279620532147633317227312, 3.60242279634992969322859654286, 4.88738845624000181590951079640, 5.22683797328678044307727658321, 6.24385332469594188633991352750, 6.54040448354097979665897131091, 7.35988039820324632836249891454, 8.020341811852842834617252662091, 8.374432549960790948234631982260, 9.425374975517169057412823621140, 9.712297620810837928519282221089, 10.36121985185948969219155767194, 11.28625091630429291855988155624, 11.42103550288830668724248085958, 12.03463020386680539693085394798, 12.20709965402453296629993409195, 13.54420653410565103323398183751

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