Properties

Label 4-140e2-1.1-c1e2-0-5
Degree $4$
Conductor $19600$
Sign $1$
Analytic cond. $1.24971$
Root an. cond. $1.05731$
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  + 2·5-s + 6·9-s − 8·19-s − 25-s − 4·29-s − 16·31-s + 12·41-s + 12·45-s − 49-s + 8·59-s − 12·61-s + 24·71-s + 8·79-s + 27·81-s + 20·89-s − 16·95-s − 36·101-s − 28·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.894·5-s + 2·9-s − 1.83·19-s − 1/5·25-s − 0.742·29-s − 2.87·31-s + 1.87·41-s + 1.78·45-s − 1/7·49-s + 1.04·59-s − 1.53·61-s + 2.84·71-s + 0.900·79-s + 3·81-s + 2.11·89-s − 1.64·95-s − 3.58·101-s − 2.68·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.24971\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{140} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420211874\)
\(L(\frac12)\) \(\approx\) \(1.420211874\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−13.19936138384739914382352658115, −12.96669995256079151075404019677, −12.56486098484899281556676128666, −12.18572119518809602416269541520, −11.10031189249695439529331808755, −10.79324829091812215006432572837, −10.45585803066577920268658596337, −9.732895421012609469365159122906, −9.187506760102842660251056287206, −9.172468262081889166146317305118, −7.85347736771192485577263380643, −7.73105409135580991251498714852, −6.70175068557014595121269523456, −6.62707383899723269380668387610, −5.65226771146563682921522801148, −5.12471343564457321137671406627, −4.03744434739196994906888648472, −3.93490860985937793236879916016, −2.27423480707383389006180586212, −1.67830909819770971279465816738, 1.67830909819770971279465816738, 2.27423480707383389006180586212, 3.93490860985937793236879916016, 4.03744434739196994906888648472, 5.12471343564457321137671406627, 5.65226771146563682921522801148, 6.62707383899723269380668387610, 6.70175068557014595121269523456, 7.73105409135580991251498714852, 7.85347736771192485577263380643, 9.172468262081889166146317305118, 9.187506760102842660251056287206, 9.732895421012609469365159122906, 10.45585803066577920268658596337, 10.79324829091812215006432572837, 11.10031189249695439529331808755, 12.18572119518809602416269541520, 12.56486098484899281556676128666, 12.96669995256079151075404019677, 13.19936138384739914382352658115

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