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Properties

Label 4-1200e2-1.1-c1e2-0-35
Degree $4$
Conductor $1440000$
Sign $-1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 10·29-s − 4·41-s − 10·49-s − 4·61-s + 30·73-s + 81-s + 10·97-s + 14·101-s − 20·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 1/3·9-s − 1.85·29-s − 0.624·41-s − 1.42·49-s − 0.512·61-s + 3.51·73-s + 1/9·81-s + 1.01·97-s + 1.39·101-s − 1.88·113-s + 2/11·121-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 + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(91.8156\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.k_cg
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.31.a_s
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.e_ao
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.43.a_k
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.47.a_acs
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.59.a_ac
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.67.a_aby
71$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.71.a_cg
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.73.abe_og
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ak_go
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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

−7.78808775913876392933453944191, −7.34130012115956754660730841468, −6.79392422706709871022858223183, −6.47014968768964921619465089530, −6.03960564462175403995142645586, −5.42134507473090919912970113015, −5.09965894321624823775818464804, −4.65053602780328874287633574450, −3.99020078596429202652449698467, −3.58555167784490939101626526313, −3.17455296312198934849494330624, −2.29734272480366233057797002406, −1.90158426083300866061178865345, −1.10993998515854918944095813467, 0, 1.10993998515854918944095813467, 1.90158426083300866061178865345, 2.29734272480366233057797002406, 3.17455296312198934849494330624, 3.58555167784490939101626526313, 3.99020078596429202652449698467, 4.65053602780328874287633574450, 5.09965894321624823775818464804, 5.42134507473090919912970113015, 6.03960564462175403995142645586, 6.47014968768964921619465089530, 6.79392422706709871022858223183, 7.34130012115956754660730841468, 7.78808775913876392933453944191

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