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Properties

Label 4-1512e2-1.1-c1e2-0-20
Degree $4$
Conductor $2286144$
Sign $-1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
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  + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 7·11-s − 12-s − 16-s + 17-s + 18-s − 7·22-s − 3·24-s + 25-s + 27-s + 5·32-s − 7·33-s + 34-s − 36-s + 8·41-s + 7·44-s − 48-s − 49-s + 50-s + 51-s + 54-s + 8·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 2.11·11-s − 0.288·12-s − 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.49·22-s − 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.883·32-s − 1.21·33-s + 0.171·34-s − 1/6·36-s + 1.24·41-s + 1.05·44-s − 0.144·48-s − 1/7·49-s + 0.141·50-s + 0.140·51-s + 0.136·54-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 - T \)
7$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.h_bi
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.13.a_ar
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.17.ab_bg
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.29.a_bl
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.37.a_aw
41$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.41.ai_dt
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.47.a_bl
53$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \) 2.53.a_dx
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ai_eo
61$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.61.a_abl
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.am_gk
71$C_2^2$ \( 1 + 113 T^{2} + p^{2} T^{4} \) 2.71.a_ej
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.u_jq
89$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.ad_gm
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.ba_nq
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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

−7.55021340121138139132028172788, −7.12093287722934697812257800065, −6.64408279202591806631224007136, −6.00130689745675468236601126086, −5.65912839458548137231722269707, −5.24347426014925060496472031583, −4.98192424248645465034998801518, −4.30150741339829332831286846124, −4.09231992074280376353867333106, −3.39299907869281763139826260832, −2.88362652028864610258781887677, −2.63460538111605865939671375325, −2.00472582769343319408708032979, −0.949983554041536032821852697187, 0, 0.949983554041536032821852697187, 2.00472582769343319408708032979, 2.63460538111605865939671375325, 2.88362652028864610258781887677, 3.39299907869281763139826260832, 4.09231992074280376353867333106, 4.30150741339829332831286846124, 4.98192424248645465034998801518, 5.24347426014925060496472031583, 5.65912839458548137231722269707, 6.00130689745675468236601126086, 6.64408279202591806631224007136, 7.12093287722934697812257800065, 7.55021340121138139132028172788

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