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Properties

Label 4-89856-1.1-c1e2-0-12
Degree $4$
Conductor $89856$
Sign $-1$
Analytic cond. $5.72929$
Root an. cond. $1.54712$
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 + 9-s − 4·11-s + 13-s − 4·23-s − 2·25-s − 27-s + 4·33-s − 8·37-s − 39-s − 12·47-s − 6·49-s − 8·59-s + 12·61-s + 4·69-s + 8·71-s − 8·73-s + 2·75-s + 81-s − 8·83-s + 24·97-s − 4·99-s + 12·107-s − 20·109-s + 8·111-s + 117-s − 6·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.834·23-s − 2/5·25-s − 0.192·27-s + 0.696·33-s − 1.31·37-s − 0.160·39-s − 1.75·47-s − 6/7·49-s − 1.04·59-s + 1.53·61-s + 0.481·69-s + 0.949·71-s − 0.936·73-s + 0.230·75-s + 1/9·81-s − 0.878·83-s + 2.43·97-s − 0.402·99-s + 1.16·107-s − 1.91·109-s + 0.759·111-s + 0.0924·117-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(89856\)    =    \(2^{8} \cdot 3^{3} \cdot 13\)
Sign: $-1$
Analytic conductor: \(5.72929\)
Root analytic conductor: \(1.54712\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 89856,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.e_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.e_bu
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.31.a_g
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.i_di
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.43.a_g
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.m_dq
53$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.53.a_adu
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
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$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.i_gc
79$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.79.a_as
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.i_eo
89$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \) 2.89.a_fy
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.97.ay_lq
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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

−9.630773043025052549118749254033, −8.817720436856553693417820440774, −8.375842125639014224692584396553, −7.88299170862236547966715942853, −7.45481252000259577781919481229, −6.80627498352812978951834952107, −6.26649543766524853241958299621, −5.80712979259510982592596900238, −5.07293394663360022989847259209, −4.88018735679668155964200289484, −3.93128434702078638217833802407, −3.36264025182393059107582573438, −2.45908743029703714512755553991, −1.59523218968436278511442735336, 0, 1.59523218968436278511442735336, 2.45908743029703714512755553991, 3.36264025182393059107582573438, 3.93128434702078638217833802407, 4.88018735679668155964200289484, 5.07293394663360022989847259209, 5.80712979259510982592596900238, 6.26649543766524853241958299621, 6.80627498352812978951834952107, 7.45481252000259577781919481229, 7.88299170862236547966715942853, 8.375842125639014224692584396553, 8.817720436856553693417820440774, 9.630773043025052549118749254033

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