Properties

Label 4-280e2-1.1-c1e2-0-20
Degree $4$
Conductor $78400$
Sign $1$
Analytic cond. $4.99885$
Root an. cond. $1.49526$
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  + 4·5-s + 5·9-s − 2·11-s + 8·19-s + 11·25-s + 2·29-s − 12·31-s − 20·41-s + 20·45-s − 49-s − 8·55-s − 12·59-s − 8·61-s − 32·71-s + 22·79-s + 16·81-s − 24·89-s + 32·95-s − 10·99-s + 30·109-s − 19·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  + 1.78·5-s + 5/3·9-s − 0.603·11-s + 1.83·19-s + 11/5·25-s + 0.371·29-s − 2.15·31-s − 3.12·41-s + 2.98·45-s − 1/7·49-s − 1.07·55-s − 1.56·59-s − 1.02·61-s − 3.79·71-s + 2.47·79-s + 16/9·81-s − 2.54·89-s + 3.28·95-s − 1.00·99-s + 2.87·109-s − 1.72·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.99885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 78400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.278132001\)
\(L(\frac12)\) \(\approx\) \(2.278132001\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 167 T^{2} + p^{2} T^{4} \)
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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

−12.01271553225125365392271835075, −11.86472219603563750029880608609, −10.73987651498033249879184501422, −10.73931340711509041282644396758, −9.973291072586787671151137302449, −9.951796029951577445884938068870, −9.282508570175823840734942219681, −9.081778348971112441835724859090, −8.263117211754458812509823761451, −7.59611645296750645328053040871, −6.97649506921161364227662129274, −6.95914517467456206402687332422, −5.92734712435679436376224320250, −5.66337032543459096211783332516, −4.92901788565123717578006711490, −4.66421356818882351463385575940, −3.49582169365356766387530324878, −2.98119450553070394676547750746, −1.77520958075780271457567887480, −1.53107027614785989238555584190, 1.53107027614785989238555584190, 1.77520958075780271457567887480, 2.98119450553070394676547750746, 3.49582169365356766387530324878, 4.66421356818882351463385575940, 4.92901788565123717578006711490, 5.66337032543459096211783332516, 5.92734712435679436376224320250, 6.95914517467456206402687332422, 6.97649506921161364227662129274, 7.59611645296750645328053040871, 8.263117211754458812509823761451, 9.081778348971112441835724859090, 9.282508570175823840734942219681, 9.951796029951577445884938068870, 9.973291072586787671151137302449, 10.73931340711509041282644396758, 10.73987651498033249879184501422, 11.86472219603563750029880608609, 12.01271553225125365392271835075

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