Properties

Label 4-48e2-1.1-c2e2-0-1
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $1.71061$
Root an. cond. $1.14363$
Motivic weight $2$
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  + 12·5-s − 3·9-s − 28·13-s − 12·17-s + 58·25-s + 60·29-s + 52·37-s − 108·41-s − 36·45-s + 50·49-s − 36·53-s − 140·61-s − 336·65-s + 164·73-s + 9·81-s − 144·85-s + 228·89-s + 68·97-s − 36·101-s + 68·109-s − 156·113-s + 84·117-s − 190·121-s − 36·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 12/5·5-s − 1/3·9-s − 2.15·13-s − 0.705·17-s + 2.31·25-s + 2.06·29-s + 1.40·37-s − 2.63·41-s − 4/5·45-s + 1.02·49-s − 0.679·53-s − 2.29·61-s − 5.16·65-s + 2.24·73-s + 1/9·81-s − 1.69·85-s + 2.56·89-s + 0.701·97-s − 0.356·101-s + 0.623·109-s − 1.38·113-s + 0.717·117-s − 1.57·121-s − 0.287·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1.71061\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.529901122\)
\(L(\frac12)\) \(\approx\) \(1.529901122\)
\(L(2)\) 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 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 190 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 674 T^{2} + p^{4} T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1490 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 54 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3266 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 2690 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6530 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 70 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 4894 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 3170 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 6674 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 13346 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 114 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{2} \)
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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

−15.55805893655081970887031122188, −15.07500885625071748659671228376, −14.32956604717959136396154646130, −14.05566349905762346618557539727, −13.45785257766367994033987543481, −13.11797727171950896711337448925, −12.11142725612442185527413440496, −11.97197820720414649674190343116, −10.68013238286750354977213958990, −10.30235033541122667450196192061, −9.586859779110616616445593412586, −9.464515365343886291359577295796, −8.538657292628325343749570992944, −7.60205019610170951882175267444, −6.57443135448348026528823162544, −6.23404957279477630116313023620, −5.20828591504842669058027768061, −4.77526890252897284178461225069, −2.76052191293842014090323933196, −2.03879976439589166055243716567, 2.03879976439589166055243716567, 2.76052191293842014090323933196, 4.77526890252897284178461225069, 5.20828591504842669058027768061, 6.23404957279477630116313023620, 6.57443135448348026528823162544, 7.60205019610170951882175267444, 8.538657292628325343749570992944, 9.464515365343886291359577295796, 9.586859779110616616445593412586, 10.30235033541122667450196192061, 10.68013238286750354977213958990, 11.97197820720414649674190343116, 12.11142725612442185527413440496, 13.11797727171950896711337448925, 13.45785257766367994033987543481, 14.05566349905762346618557539727, 14.32956604717959136396154646130, 15.07500885625071748659671228376, 15.55805893655081970887031122188

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