Properties

Label 4-393984-1.1-c1e2-0-24
Degree $4$
Conductor $393984$
Sign $-1$
Analytic cond. $25.1207$
Root an. cond. $2.23876$
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  + 2·7-s − 2·13-s − 3·19-s − 4·25-s − 10·31-s + 4·37-s − 4·43-s − 2·49-s − 8·61-s + 2·67-s + 4·73-s − 4·79-s − 4·91-s − 14·97-s + 8·103-s − 14·109-s + 14·121-s + 127-s + 131-s − 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.554·13-s − 0.688·19-s − 4/5·25-s − 1.79·31-s + 0.657·37-s − 0.609·43-s − 2/7·49-s − 1.02·61-s + 0.244·67-s + 0.468·73-s − 0.450·79-s − 0.419·91-s − 1.42·97-s + 0.788·103-s − 1.34·109-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 393984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(393984\)    =    \(2^{8} \cdot 3^{4} \cdot 19\)
Sign: $-1$
Analytic conductor: \(25.1207\)
Root analytic conductor: \(2.23876\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{393984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 393984,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
19$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−8.436014133954077079859429713470, −7.972061565144407306239724725950, −7.50082145785040862284882032540, −7.18556086256179291328931774455, −6.59614308797099970843070229545, −5.99398805824249565656264630191, −5.65099554200778464175637378520, −4.94990709256801969583237259726, −4.69185485391124749926776000198, −3.96344028670972854021940739595, −3.54747462139639630065374678945, −2.67621176389221039113599879534, −2.06113623584082155239802235407, −1.42953597992304580656157790271, 0, 1.42953597992304580656157790271, 2.06113623584082155239802235407, 2.67621176389221039113599879534, 3.54747462139639630065374678945, 3.96344028670972854021940739595, 4.69185485391124749926776000198, 4.94990709256801969583237259726, 5.65099554200778464175637378520, 5.99398805824249565656264630191, 6.59614308797099970843070229545, 7.18556086256179291328931774455, 7.50082145785040862284882032540, 7.972061565144407306239724725950, 8.436014133954077079859429713470

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