Properties

Label 4-39e4-1.1-c1e2-0-5
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  + 4·4-s + 12·16-s + 12·23-s − 2·25-s − 12·29-s − 2·43-s + 11·49-s − 24·53-s + 2·61-s + 32·64-s − 22·79-s + 48·92-s − 8·100-s + 36·101-s − 2·103-s + 12·107-s − 12·113-s − 48·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2·4-s + 3·16-s + 2.50·23-s − 2/5·25-s − 2.22·29-s − 0.304·43-s + 11/7·49-s − 3.29·53-s + 0.256·61-s + 4·64-s − 2.47·79-s + 5.00·92-s − 4/5·100-s + 3.58·101-s − 0.197·103-s + 1.16·107-s − 1.12·113-s − 4.45·116-s + 0.909·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.323572361\)
\(L(\frac12)\) \(\approx\) \(4.323572361\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−9.840291780894637970062636974721, −9.102409662652452915767317586732, −9.035578530809179338641130677321, −8.430514596693369351002158169863, −7.72262467738625320139130400839, −7.68413198227570546914699010292, −7.22821951012821342910087383198, −6.86764579918031861030213549245, −6.58144726406100902676289329665, −5.88536566003845186685257809154, −5.80923734757000170175100448326, −5.25056178523445820072042224915, −4.74812219646033705570892151709, −4.09028629176335646176633137232, −3.32399821595725790197619322353, −3.19996939244799445412646580023, −2.69014699240501934300604652367, −1.85160858216345864639681883169, −1.72960496393829983522528904597, −0.802733956456812466215252951307, 0.802733956456812466215252951307, 1.72960496393829983522528904597, 1.85160858216345864639681883169, 2.69014699240501934300604652367, 3.19996939244799445412646580023, 3.32399821595725790197619322353, 4.09028629176335646176633137232, 4.74812219646033705570892151709, 5.25056178523445820072042224915, 5.80923734757000170175100448326, 5.88536566003845186685257809154, 6.58144726406100902676289329665, 6.86764579918031861030213549245, 7.22821951012821342910087383198, 7.68413198227570546914699010292, 7.72262467738625320139130400839, 8.430514596693369351002158169863, 9.035578530809179338641130677321, 9.102409662652452915767317586732, 9.840291780894637970062636974721

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