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Properties

Label 4-450e2-1.1-c1e2-0-8
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

\(L(1)\) \(\approx\) \(2.102611948\)
\(L(\frac12)\) \(\approx\) \(2.102611948\)
\(L(\frac{3}{2})\) 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.am_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.am_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.m_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.y_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.y_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

−11.41259106026623354709253200328, −11.04238032527484228369273696027, −10.16572437153849807630074510989, −10.00527370541858331902712317373, −9.261949268368779885007966893660, −9.172244035568603665863444619636, −8.822788090344997879108240222199, −8.327062825083179742073637693113, −7.42803712387726107542782245603, −7.27066054679861735713409817004, −6.52060427724931337192128274937, −6.35378265604245827247284872400, −5.67061075796970096100998333241, −5.08681591364366779099800787329, −4.30730624874092149814131848953, −4.04883168171756887383131827482, −3.47831764970121543239869758483, −2.82189013550091374750763219830, −1.40514541273076448597784095033, −1.19207266165402611601661331382, 1.19207266165402611601661331382, 1.40514541273076448597784095033, 2.82189013550091374750763219830, 3.47831764970121543239869758483, 4.04883168171756887383131827482, 4.30730624874092149814131848953, 5.08681591364366779099800787329, 5.67061075796970096100998333241, 6.35378265604245827247284872400, 6.52060427724931337192128274937, 7.27066054679861735713409817004, 7.42803712387726107542782245603, 8.327062825083179742073637693113, 8.822788090344997879108240222199, 9.172244035568603665863444619636, 9.261949268368779885007966893660, 10.00527370541858331902712317373, 10.16572437153849807630074510989, 11.04238032527484228369273696027, 11.41259106026623354709253200328

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