Properties

Label 4-1400e2-1.1-c1e2-0-4
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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 − 16·19-s − 12·29-s + 16·31-s + 4·41-s − 49-s − 12·61-s − 16·71-s − 32·79-s + 27·81-s + 12·89-s − 48·99-s + 4·101-s + 20·109-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 − 96·171-s + ⋯
L(s)  = 1  + 2·9-s − 2.41·11-s − 3.67·19-s − 2.22·29-s + 2.87·31-s + 0.624·41-s − 1/7·49-s − 1.53·61-s − 1.89·71-s − 3.60·79-s + 3·81-s + 1.27·89-s − 4.82·99-s + 0.398·101-s + 1.91·109-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 − 7.34·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.054517411\)
\(L(\frac12)\) \(\approx\) \(1.054517411\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−9.992903094702103705351172268204, −9.410916468241825229766145265603, −8.919330382079513576548056115159, −8.396385774894016392694733449916, −8.243724306143163975356336755463, −7.62987080601243796752423280141, −7.47011285572364215902889192047, −7.00781810551954968235323917709, −6.47758094124440516270583930495, −6.02224124161427072590741799890, −5.81206519022109189319313440131, −4.94747425022579641938684617095, −4.61039410653294145651867066362, −4.32377305846473301833053232353, −4.01828761299910274334162031536, −3.08334817946026054783807946778, −2.59723362413832936647261406193, −2.03056861323365657171738892390, −1.64670023248576994942418186967, −0.39778913001841796688175152024, 0.39778913001841796688175152024, 1.64670023248576994942418186967, 2.03056861323365657171738892390, 2.59723362413832936647261406193, 3.08334817946026054783807946778, 4.01828761299910274334162031536, 4.32377305846473301833053232353, 4.61039410653294145651867066362, 4.94747425022579641938684617095, 5.81206519022109189319313440131, 6.02224124161427072590741799890, 6.47758094124440516270583930495, 7.00781810551954968235323917709, 7.47011285572364215902889192047, 7.62987080601243796752423280141, 8.243724306143163975356336755463, 8.396385774894016392694733449916, 8.919330382079513576548056115159, 9.410916468241825229766145265603, 9.992903094702103705351172268204

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