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Properties

Label 4-450e2-1.1-c1e2-0-2
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 12·11-s + 16-s + 8·19-s − 12·29-s − 8·31-s + 12·44-s + 10·49-s + 12·59-s + 4·61-s − 64-s + 24·71-s − 8·76-s + 8·79-s + 24·89-s + 12·101-s − 4·109-s + 12·116-s + 86·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s − 3.61·11-s + 1/4·16-s + 1.83·19-s − 2.22·29-s − 1.43·31-s + 1.80·44-s + 10/7·49-s + 1.56·59-s + 0.512·61-s − 1/8·64-s + 2.84·71-s − 0.917·76-s + 0.900·79-s + 2.54·89-s + 1.19·101-s − 0.383·109-s + 1.11·116-s + 7.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8068376040\)
\(L(\frac12)\) \(\approx\) \(0.8068376040\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−11.46691306363255200295706673721, −10.71188889806956546895894555819, −10.46532617918269603033219123694, −10.02594060123778889785594044970, −9.550681759427057251586894441137, −9.146893401147306246525516267866, −8.588281466561764313932736470777, −7.87067540107579354916680303729, −7.73966030085716759971920989634, −7.45493573375307987980438699169, −6.86972012736796772919812145513, −5.72241631195504576264873927835, −5.59588701731174473588947707537, −5.09655250512672301948311836631, −4.93617897661492578500048103431, −3.68840496646488921396534688276, −3.48592334877177566588123226046, −2.51697367074953278995786915732, −2.15003086525239329527872874140, −0.54501603361966242488037523888, 0.54501603361966242488037523888, 2.15003086525239329527872874140, 2.51697367074953278995786915732, 3.48592334877177566588123226046, 3.68840496646488921396534688276, 4.93617897661492578500048103431, 5.09655250512672301948311836631, 5.59588701731174473588947707537, 5.72241631195504576264873927835, 6.86972012736796772919812145513, 7.45493573375307987980438699169, 7.73966030085716759971920989634, 7.87067540107579354916680303729, 8.588281466561764313932736470777, 9.146893401147306246525516267866, 9.550681759427057251586894441137, 10.02594060123778889785594044970, 10.46532617918269603033219123694, 10.71188889806956546895894555819, 11.46691306363255200295706673721

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