Properties

Label 4-1778112-1.1-c1e2-0-2
Degree $4$
Conductor $1778112$
Sign $1$
Analytic cond. $113.373$
Root an. cond. $3.26308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 7-s + 3·8-s − 14-s − 16-s + 6·25-s − 28-s − 5·32-s + 49-s − 6·50-s + 3·56-s + 7·64-s + 32·79-s − 98-s − 6·100-s − 112-s − 28·113-s − 6·121-s + 127-s + 3·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 0.267·14-s − 1/4·16-s + 6/5·25-s − 0.188·28-s − 0.883·32-s + 1/7·49-s − 0.848·50-s + 0.400·56-s + 7/8·64-s + 3.60·79-s − 0.101·98-s − 3/5·100-s − 0.0944·112-s − 2.63·113-s − 0.545·121-s + 0.0887·127-s + 0.265·128-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1778112\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(113.373\)
Root analytic conductor: \(3.26308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1778112,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.228798671\)
\(L(\frac12)\) \(\approx\) \(1.228798671\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3 \( 1 \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.030514768084889787852043098059, −7.48733483740196396053226289227, −7.06594028263065496300867747391, −6.63478496161881034331199213050, −6.22628169315218824079820526079, −5.51186257816528055742974819242, −5.18431446031001017815063505262, −4.78405858770322399321481635281, −4.31510861551258387176284642505, −3.79891101220150690008710647647, −3.28648042377121578295375043576, −2.58732362847431751508943243155, −1.95450748716524399321738996754, −1.26168385839125974576745925039, −0.58503832335070089164870164959, 0.58503832335070089164870164959, 1.26168385839125974576745925039, 1.95450748716524399321738996754, 2.58732362847431751508943243155, 3.28648042377121578295375043576, 3.79891101220150690008710647647, 4.31510861551258387176284642505, 4.78405858770322399321481635281, 5.18431446031001017815063505262, 5.51186257816528055742974819242, 6.22628169315218824079820526079, 6.63478496161881034331199213050, 7.06594028263065496300867747391, 7.48733483740196396053226289227, 8.030514768084889787852043098059

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