Properties

Label 4-20e4-1.1-c1e2-0-3
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $10.2017$
Root an. cond. $1.78718$
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  + 6·9-s − 8·11-s + 8·19-s + 4·29-s + 16·31-s − 12·41-s − 2·49-s − 8·59-s − 4·61-s + 27·81-s + 12·89-s − 48·99-s + 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 48·171-s + 173-s + ⋯
L(s)  = 1  + 2·9-s − 2.41·11-s + 1.83·19-s + 0.742·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s − 1.04·59-s − 0.512·61-s + 3·81-s + 1.27·89-s − 4.82·99-s + 1.19·101-s − 2.68·109-s + 2.36·121-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 + 1.69·169-s + 3.67·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(10.2017\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.762784313\)
\(L(\frac12)\) \(\approx\) \(1.762784313\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−11.66637468486795400945844347698, −10.77212921437032853485016691296, −10.36371787716092953955720282987, −10.34432295746027101544622215853, −9.687941945913011033985574344810, −9.569900651218649872082119047581, −8.669159741380960439502819043824, −8.019356680266294677456073115161, −7.82345485548024244995230582501, −7.49266168316699170013362263022, −6.69750607834347011257043200244, −6.56463226477988062593447218137, −5.57782202616065436043968572110, −5.08020078559631303189439793818, −4.74860444575389548886244759317, −4.22963467981581805606017116424, −3.16295329624880242409614353354, −2.89662801195393895582887534117, −1.89590827212659631097713103959, −0.939278969634963650164151672092, 0.939278969634963650164151672092, 1.89590827212659631097713103959, 2.89662801195393895582887534117, 3.16295329624880242409614353354, 4.22963467981581805606017116424, 4.74860444575389548886244759317, 5.08020078559631303189439793818, 5.57782202616065436043968572110, 6.56463226477988062593447218137, 6.69750607834347011257043200244, 7.49266168316699170013362263022, 7.82345485548024244995230582501, 8.019356680266294677456073115161, 8.669159741380960439502819043824, 9.569900651218649872082119047581, 9.687941945913011033985574344810, 10.34432295746027101544622215853, 10.36371787716092953955720282987, 10.77212921437032853485016691296, 11.66637468486795400945844347698

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