Properties

Label 4-1150e2-1.1-c1e2-0-5
Degree $4$
Conductor $1322500$
Sign $1$
Analytic cond. $84.3237$
Root an. cond. $3.03031$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 3·4-s + 2·6-s − 7-s + 4·8-s + 3·11-s + 3·12-s − 7·13-s − 2·14-s + 5·16-s + 3·17-s + 7·19-s − 21-s + 6·22-s − 2·23-s + 4·24-s − 14·26-s + 2·27-s − 3·28-s + 6·29-s + 7·31-s + 6·32-s + 3·33-s + 6·34-s + 8·37-s + 14·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s − 0.377·7-s + 1.41·8-s + 0.904·11-s + 0.866·12-s − 1.94·13-s − 0.534·14-s + 5/4·16-s + 0.727·17-s + 1.60·19-s − 0.218·21-s + 1.27·22-s − 0.417·23-s + 0.816·24-s − 2.74·26-s + 0.384·27-s − 0.566·28-s + 1.11·29-s + 1.25·31-s + 1.06·32-s + 0.522·33-s + 1.02·34-s + 1.31·37-s + 2.27·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1322500\)    =    \(2^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(84.3237\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1150} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1322500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.068182284\)
\(L(\frac12)\) \(\approx\) \(7.068182284\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 18 T + 154 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−9.901638178568965617970020386986, −9.747972690496255641845605801117, −9.235552577769392690316669871255, −8.940352463843414538479869514280, −7.957169036281539429774128180976, −7.912786698904494383053855968820, −7.60807321389107926086352318157, −6.89794079317323013227175124374, −6.63688729260059636407148142703, −6.26098428855000817717282304546, −5.71552343620791958133274735820, −5.15515465410233597640554913891, −4.69270388701496293375701357770, −4.60694076861134357901486980124, −3.71896873462567333673682235845, −3.39351292271383808193163982811, −2.72054382036392625050348153692, −2.68347578616787567756974686316, −1.73086645546949651249292237371, −0.926352951191884097594504054965, 0.926352951191884097594504054965, 1.73086645546949651249292237371, 2.68347578616787567756974686316, 2.72054382036392625050348153692, 3.39351292271383808193163982811, 3.71896873462567333673682235845, 4.60694076861134357901486980124, 4.69270388701496293375701357770, 5.15515465410233597640554913891, 5.71552343620791958133274735820, 6.26098428855000817717282304546, 6.63688729260059636407148142703, 6.89794079317323013227175124374, 7.60807321389107926086352318157, 7.912786698904494383053855968820, 7.957169036281539429774128180976, 8.940352463843414538479869514280, 9.235552577769392690316669871255, 9.747972690496255641845605801117, 9.901638178568965617970020386986

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