Properties

Label 4-24e3-1.1-c1e2-0-1
Degree $4$
Conductor $13824$
Sign $1$
Analytic cond. $0.881430$
Root an. cond. $0.968940$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 4·13-s + 8·23-s + 2·25-s − 27-s − 4·37-s − 4·39-s + 8·47-s + 2·49-s − 24·59-s − 4·61-s − 8·69-s + 8·71-s − 12·73-s − 2·75-s + 81-s + 16·83-s − 12·97-s − 8·107-s − 12·109-s + 4·111-s + 4·117-s − 6·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.10·13-s + 1.66·23-s + 2/5·25-s − 0.192·27-s − 0.657·37-s − 0.640·39-s + 1.16·47-s + 2/7·49-s − 3.12·59-s − 0.512·61-s − 0.963·69-s + 0.949·71-s − 1.40·73-s − 0.230·75-s + 1/9·81-s + 1.75·83-s − 1.21·97-s − 0.773·107-s − 1.14·109-s + 0.379·111-s + 0.369·117-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13824\)    =    \(2^{9} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.881430\)
Root analytic conductor: \(0.968940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9414438654\)
\(L(\frac12)\) \(\approx\) \(0.9414438654\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ae_o
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ai_bu
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.29.a_ac
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.e_o
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.41.a_aco
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ai_dq
53$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.53.a_da
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.67.a_cc
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.83.aq_ig
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.89.a_ade
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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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

−10.98649795827175662896495645062, −10.80270779125254988619827761612, −10.39858023834777749202821229548, −9.384397161262102629236538233716, −9.124893136631547282281540998646, −8.491810722799401088606502239098, −7.75956031279287530663715100088, −7.14236645452677075874292147631, −6.52441673563482982505263882438, −5.96674617007111413602155038675, −5.24063077562821841958336081579, −4.60686286877679965426088692543, −3.72469107346195531692365458658, −2.86061973969156559039441394196, −1.34132199587932372331895421307, 1.34132199587932372331895421307, 2.86061973969156559039441394196, 3.72469107346195531692365458658, 4.60686286877679965426088692543, 5.24063077562821841958336081579, 5.96674617007111413602155038675, 6.52441673563482982505263882438, 7.14236645452677075874292147631, 7.75956031279287530663715100088, 8.491810722799401088606502239098, 9.124893136631547282281540998646, 9.384397161262102629236538233716, 10.39858023834777749202821229548, 10.80270779125254988619827761612, 10.98649795827175662896495645062

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