Properties

Label 4-117e2-1.1-c3e2-0-7
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $47.6544$
Root an. cond. $2.62739$
Motivic weight $3$
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  + 16·4-s + 70·13-s + 192·16-s + 250·25-s − 1.04e3·43-s − 286·49-s + 1.12e3·52-s − 364·61-s + 2.04e3·64-s − 1.76e3·79-s + 4.00e3·100-s + 3.64e3·103-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.70e3·169-s − 1.66e4·172-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·4-s + 1.49·13-s + 3·16-s + 2·25-s − 3.68·43-s − 0.833·49-s + 2.98·52-s − 0.764·61-s + 4·64-s − 2.51·79-s + 4·100-s + 3.48·103-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.23·169-s − 7.37·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(47.6544\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.992252686\)
\(L(\frac12)\) \(\approx\) \(3.992252686\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - 70 T + p^{3} T^{2} \)
good2$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 20 T + p^{3} T^{2} ) \)
11$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 56 T + p^{3} T^{2} ) \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \)
37$C_2$ \( ( 1 - 110 T + p^{3} T^{2} )( 1 + 110 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 520 T + p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 182 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 880 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 1190 T + p^{3} T^{2} )( 1 + 1190 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 + 884 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \)
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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

−13.09793682249852029627757446505, −12.81104306052950314221631167958, −12.15207452153091951168640441358, −11.60737451788467510518812776262, −11.22253296543019460382726301899, −10.92732715173298755552201689528, −10.13361477787635338604374478398, −10.05881747090315228572785022402, −8.767159425908710069810667880785, −8.500916034879971704384469377688, −7.80401112502456303824670417537, −7.07854376181715442610251502412, −6.65201975302685759069487382070, −6.22795978536921027379837434826, −5.54771547783922769648659440040, −4.68619552344503991237305000032, −3.28794952208295891947377017488, −3.19344125700380133116637973287, −1.92033501367808232448959916594, −1.20959907316659183120245590820, 1.20959907316659183120245590820, 1.92033501367808232448959916594, 3.19344125700380133116637973287, 3.28794952208295891947377017488, 4.68619552344503991237305000032, 5.54771547783922769648659440040, 6.22795978536921027379837434826, 6.65201975302685759069487382070, 7.07854376181715442610251502412, 7.80401112502456303824670417537, 8.500916034879971704384469377688, 8.767159425908710069810667880785, 10.05881747090315228572785022402, 10.13361477787635338604374478398, 10.92732715173298755552201689528, 11.22253296543019460382726301899, 11.60737451788467510518812776262, 12.15207452153091951168640441358, 12.81104306052950314221631167958, 13.09793682249852029627757446505

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