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Properties

Label 4-504e2-1.1-c1e2-0-30
Degree $4$
Conductor $254016$
Sign $-1$
Analytic cond. $16.1962$
Root an. cond. $2.00610$
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  − 2·4-s − 5·7-s + 4·16-s − 5·25-s + 10·28-s + 15·31-s + 18·49-s − 8·64-s − 13·79-s − 24·97-s + 10·100-s − 20·112-s + 11·121-s − 30·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 173-s + 25·175-s + 179-s + ⋯
L(s)  = 1  − 4-s − 1.88·7-s + 16-s − 25-s + 1.88·28-s + 2.69·31-s + 18/7·49-s − 64-s − 1.46·79-s − 2.43·97-s + 100-s − 1.88·112-s + 121-s − 2.69·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s + 0.0760·173-s + 1.88·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(16.1962\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 254016,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.5.a_f
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.19.a_l
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.ap_ec
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.a_cv
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.53.a_acb
59$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.59.a_ch
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.61.a_cw
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.a_n
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.a_dt
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.n_dm
83$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.83.a_df
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.y_ld
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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

−8.733858458046541620799844399607, −8.287622791972123051114421336375, −7.930467096203427916021314282505, −7.15125647105771756652521784477, −6.83873180984100009834354784283, −6.10425495264395646457310150796, −5.97692474895246488120121812341, −5.34224536567492353331325749057, −4.53854225341309252590966546950, −4.22260366715298085745501835392, −3.53069709885924789345147110252, −3.06041999573767857587752167246, −2.43293902032096485806058479120, −1.06452962022324606436744164592, 0, 1.06452962022324606436744164592, 2.43293902032096485806058479120, 3.06041999573767857587752167246, 3.53069709885924789345147110252, 4.22260366715298085745501835392, 4.53854225341309252590966546950, 5.34224536567492353331325749057, 5.97692474895246488120121812341, 6.10425495264395646457310150796, 6.83873180984100009834354784283, 7.15125647105771756652521784477, 7.930467096203427916021314282505, 8.287622791972123051114421336375, 8.733858458046541620799844399607

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