Properties

Label 4-48e2-1.1-c12e2-0-0
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $1924.72$
Root an. cond. $6.62357$
Motivic weight $12$
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  + 1.35e3·3-s − 8.05e4·7-s + 1.29e6·9-s + 2.56e6·13-s − 1.06e8·19-s − 1.08e8·21-s + 4.26e7·25-s + 1.02e9·27-s − 1.33e8·31-s + 4.45e9·37-s + 3.46e9·39-s − 1.79e10·43-s − 2.28e10·49-s − 1.44e11·57-s − 8.13e10·61-s − 1.03e11·63-s − 2.42e11·67-s − 1.21e11·73-s + 5.75e10·75-s + 5.04e11·79-s + 6.98e11·81-s − 2.06e11·91-s − 1.79e11·93-s + 1.30e12·97-s + 1.99e12·103-s + 5.42e12·109-s + 6.01e12·111-s + ⋯
L(s)  = 1  + 1.85·3-s − 0.684·7-s + 2.42·9-s + 0.532·13-s − 2.26·19-s − 1.26·21-s + 0.174·25-s + 2.64·27-s − 0.149·31-s + 1.73·37-s + 0.985·39-s − 2.84·43-s − 1.64·49-s − 4.19·57-s − 1.57·61-s − 1.66·63-s − 2.67·67-s − 0.805·73-s + 0.323·75-s + 2.07·79-s + 2.47·81-s − 0.364·91-s − 0.277·93-s + 1.56·97-s + 1.66·103-s + 3.23·109-s + 3.21·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1924.72\)
Root analytic conductor: \(6.62357\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(4.228485408\)
\(L(\frac12)\) \(\approx\) \(4.228485408\)
\(L(7)\) 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$C_2$ \( 1 - 50 p^{3} T + p^{12} T^{2} \)
good5$C_2^2$ \( 1 - 1706066 p^{2} T^{2} + p^{24} T^{4} \)
7$C_2$ \( ( 1 + 5750 p T + p^{12} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 40734109202 p^{2} T^{2} + p^{24} T^{4} \)
13$C_2$ \( ( 1 - 1284050 T + p^{12} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 944884986604418 T^{2} + p^{24} T^{4} \)
19$C_2$ \( ( 1 + 53343578 T + p^{12} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 61021177942562 p^{2} T^{2} + p^{24} T^{4} \)
29$C_2^2$ \( 1 - 693172036445878082 T^{2} + p^{24} T^{4} \)
31$C_2$ \( ( 1 + 66526202 T + p^{12} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 60235850 p T + p^{12} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 22304779456187067838 T^{2} + p^{24} T^{4} \)
43$C_2$ \( ( 1 + 8977216250 T + p^{12} T^{2} )^{2} \)
47$C_2^2$ \( 1 - \)\(23\!\cdots\!78\)\( T^{2} + p^{24} T^{4} \)
53$C_2^2$ \( 1 + \)\(71\!\cdots\!22\)\( T^{2} + p^{24} T^{4} \)
59$C_2^2$ \( 1 - \)\(14\!\cdots\!62\)\( T^{2} + p^{24} T^{4} \)
61$C_2$ \( ( 1 + 40679935918 T + p^{12} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 121176846650 T + p^{12} T^{2} )^{2} \)
71$C_2^2$ \( 1 - \)\(30\!\cdots\!82\)\( T^{2} + p^{24} T^{4} \)
73$C_2$ \( ( 1 + 60956187550 T + p^{12} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 252324997702 T + p^{12} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(45\!\cdots\!38\)\( T^{2} + p^{24} T^{4} \)
89$C_2^2$ \( 1 - \)\(48\!\cdots\!42\)\( T^{2} + p^{24} T^{4} \)
97$C_2$ \( ( 1 - 653817778850 T + p^{12} T^{2} )^{2} \)
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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

−13.78274096261404263966985078903, −12.94143704180348846851045826331, −12.58237027850061467834121613916, −11.62149730566501456321983094449, −10.80749802229184699385798997736, −10.15462531681076999759648517298, −9.759985367229087386909711793827, −8.803974766395477417427555345727, −8.798563574704168200585674162062, −7.972686286782029676362253172136, −7.46249598350074287634340965126, −6.36612318864778036977557023562, −6.34759229565675892627514907850, −4.70095750750286450373814188972, −4.28461975273093839152701020498, −3.29774003194586230876578422203, −3.09459075544149254503429887646, −2.02029025834869904852842509804, −1.68555126838189036953492754432, −0.47747868281282726649641745259, 0.47747868281282726649641745259, 1.68555126838189036953492754432, 2.02029025834869904852842509804, 3.09459075544149254503429887646, 3.29774003194586230876578422203, 4.28461975273093839152701020498, 4.70095750750286450373814188972, 6.34759229565675892627514907850, 6.36612318864778036977557023562, 7.46249598350074287634340965126, 7.972686286782029676362253172136, 8.798563574704168200585674162062, 8.803974766395477417427555345727, 9.759985367229087386909711793827, 10.15462531681076999759648517298, 10.80749802229184699385798997736, 11.62149730566501456321983094449, 12.58237027850061467834121613916, 12.94143704180348846851045826331, 13.78274096261404263966985078903

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