Properties

Label 4-1200e2-1.1-c1e2-0-20
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
Motivic weight $1$
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  + 3·3-s + 6·9-s − 6·11-s − 4·13-s + 12·23-s + 9·27-s − 18·33-s + 16·37-s − 12·39-s − 12·47-s + 14·49-s − 24·59-s + 16·61-s + 36·69-s + 12·71-s + 2·73-s + 9·81-s − 18·83-s + 20·97-s − 36·99-s + 6·107-s − 16·109-s + 48·111-s − 24·117-s + 5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 1.80·11-s − 1.10·13-s + 2.50·23-s + 1.73·27-s − 3.13·33-s + 2.63·37-s − 1.92·39-s − 1.75·47-s + 2·49-s − 3.12·59-s + 2.04·61-s + 4.33·69-s + 1.42·71-s + 0.234·73-s + 81-s − 1.97·83-s + 2.03·97-s − 3.61·99-s + 0.580·107-s − 1.53·109-s + 4.55·111-s − 2.21·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-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: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.690799168\)
\(L(\frac12)\) \(\approx\) \(3.690799168\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 151 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−9.949071215160877709405285288689, −9.388712944607297374453685683482, −9.148258902572960841846009259247, −8.807714535280942665412841301694, −8.255774497202089660090663603470, −7.77666664841938190662447383816, −7.71661740425551270066394270788, −7.34009431470494632789417767851, −6.77560219337028756121239169548, −6.40687562695412033412906492299, −5.54815261604095958672505092947, −5.18343743253181781866862967579, −4.76447631261619758628405568919, −4.34226208320684842695419885883, −3.71331411593505908329064024068, −2.88253801903961845956274903295, −2.85753585158968917578674971710, −2.48302961494227328366943154877, −1.70698974645208637812633518352, −0.73878341732076462904245968158, 0.73878341732076462904245968158, 1.70698974645208637812633518352, 2.48302961494227328366943154877, 2.85753585158968917578674971710, 2.88253801903961845956274903295, 3.71331411593505908329064024068, 4.34226208320684842695419885883, 4.76447631261619758628405568919, 5.18343743253181781866862967579, 5.54815261604095958672505092947, 6.40687562695412033412906492299, 6.77560219337028756121239169548, 7.34009431470494632789417767851, 7.71661740425551270066394270788, 7.77666664841938190662447383816, 8.255774497202089660090663603470, 8.807714535280942665412841301694, 9.148258902572960841846009259247, 9.388712944607297374453685683482, 9.949071215160877709405285288689

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