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Properties

Label 4-24768-1.1-c1e2-0-3
Degree $4$
Conductor $24768$
Sign $-1$
Analytic cond. $1.57922$
Root an. cond. $1.12101$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 2·9-s − 9·11-s − 2·12-s + 4·16-s − 3·17-s − 5·19-s − 25-s − 5·27-s − 9·33-s + 4·36-s + 9·41-s + 9·43-s + 18·44-s + 4·48-s − 4·49-s − 3·51-s − 5·57-s + 9·59-s − 8·64-s − 8·67-s + 6·68-s + 13·73-s − 75-s + 10·76-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 2/3·9-s − 2.71·11-s − 0.577·12-s + 16-s − 0.727·17-s − 1.14·19-s − 1/5·25-s − 0.962·27-s − 1.56·33-s + 2/3·36-s + 1.40·41-s + 1.37·43-s + 2.71·44-s + 0.577·48-s − 4/7·49-s − 0.420·51-s − 0.662·57-s + 1.17·59-s − 64-s − 0.977·67-s + 0.727·68-s + 1.52·73-s − 0.115·75-s + 1.14·76-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24768 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24768 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24768\)    =    \(2^{6} \cdot 3^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(1.57922\)
Root analytic conductor: \(1.12101\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 24768,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 8 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.j_bo
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.13.a_t
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.f_y
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.29.a_n
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.37.a_ca
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.41.aj_dw
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.aj_eo
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.61.a_bo
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.an_gm
79$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.79.a_adr
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ad_eu
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.c_ek
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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

−10.37025852540314366589865176059, −9.898528402766603937432247999617, −9.255259851466102191379249626439, −8.716131785819737139671870223010, −8.335992044099456736801834933861, −7.79143367558868198837611486094, −7.50670396924313148836146436795, −6.40800395246911910100613436882, −5.64572905896902812398524120883, −5.30006456695466704038184683110, −4.53500022519960414357345916890, −3.88805733020808846294715965165, −2.79014156144437365266887153134, −2.39865581228781781251540004010, 0, 2.39865581228781781251540004010, 2.79014156144437365266887153134, 3.88805733020808846294715965165, 4.53500022519960414357345916890, 5.30006456695466704038184683110, 5.64572905896902812398524120883, 6.40800395246911910100613436882, 7.50670396924313148836146436795, 7.79143367558868198837611486094, 8.335992044099456736801834933861, 8.716131785819737139671870223010, 9.255259851466102191379249626439, 9.898528402766603937432247999617, 10.37025852540314366589865176059

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