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Properties

Label 4-89856-1.1-c1e2-0-13
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 + 5·13-s − 12·23-s − 6·25-s − 27-s + 4·33-s − 18·37-s − 5·39-s + 10·47-s − 4·49-s + 8·59-s + 2·61-s + 12·69-s − 12·71-s + 6·73-s + 6·75-s + 81-s − 8·83-s − 12·97-s − 4·99-s + 4·107-s + 14·109-s + 18·111-s + 5·117-s + 6·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.38·13-s − 2.50·23-s − 6/5·25-s − 0.192·27-s + 0.696·33-s − 2.95·37-s − 0.800·39-s + 1.45·47-s − 4/7·49-s + 1.04·59-s + 0.256·61-s + 1.44·69-s − 1.42·71-s + 0.702·73-s + 0.692·75-s + 1/9·81-s − 0.878·83-s − 1.21·97-s − 0.402·99-s + 0.386·107-s + 1.34·109-s + 1.70·111-s + 0.462·117-s + 6/11·121-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$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.e_k
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.m_da
29$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.29.a_u
31$C_2^2$ \( 1 - 60 T^{2} + p^{2} T^{4} \) 2.31.a_aci
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.s_fq
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.ak_cs
53$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.53.a_adc
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.ac_bq
67$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.67.a_aco
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.m_fm
73$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ag_cw
79$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.79.a_aeo
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.i_gk
89$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.89.a_ck
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

−9.524811990327994550461536332441, −8.770137187982290334159011299348, −8.363767168178449551323442665303, −7.998379844258845816073274760932, −7.37497260420365839759678894085, −6.88163451054120410871295459715, −6.12772788593838726278931147857, −5.76543344645235926489805006842, −5.42273010312330363104014553154, −4.61197406367694580096900637405, −3.86505460837619360642154169006, −3.53341626513699331154563399162, −2.35643960379084741509267931613, −1.63988711670840286382739524348, 0, 1.63988711670840286382739524348, 2.35643960379084741509267931613, 3.53341626513699331154563399162, 3.86505460837619360642154169006, 4.61197406367694580096900637405, 5.42273010312330363104014553154, 5.76543344645235926489805006842, 6.12772788593838726278931147857, 6.88163451054120410871295459715, 7.37497260420365839759678894085, 7.998379844258845816073274760932, 8.363767168178449551323442665303, 8.770137187982290334159011299348, 9.524811990327994550461536332441

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