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Properties

Label 4-383328-1.1-c1e2-0-9
Degree $4$
Conductor $383328$
Sign $-1$
Analytic cond. $24.4413$
Root an. cond. $2.22346$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 8·5-s − 4·7-s + 8-s + 9-s − 8·10-s + 11-s − 4·14-s + 16-s + 18-s − 8·20-s + 22-s + 38·25-s − 4·28-s + 32-s + 32·35-s + 36-s − 4·37-s − 8·40-s + 8·43-s + 44-s − 8·45-s − 2·49-s + 38·50-s + 8·53-s − 8·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 3.57·5-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 2.52·10-s + 0.301·11-s − 1.06·14-s + 1/4·16-s + 0.235·18-s − 1.78·20-s + 0.213·22-s + 38/5·25-s − 0.755·28-s + 0.176·32-s + 5.40·35-s + 1/6·36-s − 0.657·37-s − 1.26·40-s + 1.21·43-s + 0.150·44-s − 1.19·45-s − 2/7·49-s + 5.37·50-s + 1.09·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 383328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(383328\)    =    \(2^{5} \cdot 3^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(24.4413\)
Root analytic conductor: \(2.22346\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 383328,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.5.i_ba
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.235820094350727824625666284447, −7.72339303053252541131209387915, −7.68407763492968955980629110622, −7.03737354779624543347790680337, −6.60367482888104144370679944075, −6.46917181705411800265373292912, −5.42222207046297112948173299542, −4.87437307057030940088074522867, −4.25842178547984493307347569445, −3.99255436866847289087860194041, −3.43242489959271970824287162411, −3.34833436263269527102959669443, −2.53491081695155997479876397633, −0.890028096017018980688822052293, 0, 0.890028096017018980688822052293, 2.53491081695155997479876397633, 3.34833436263269527102959669443, 3.43242489959271970824287162411, 3.99255436866847289087860194041, 4.25842178547984493307347569445, 4.87437307057030940088074522867, 5.42222207046297112948173299542, 6.46917181705411800265373292912, 6.60367482888104144370679944075, 7.03737354779624543347790680337, 7.68407763492968955980629110622, 7.72339303053252541131209387915, 8.235820094350727824625666284447

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