Properties

Label 4-13248-1.1-c1e2-0-2
Degree $4$
Conductor $13248$
Sign $-1$
Analytic cond. $0.844703$
Root an. cond. $0.958685$
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  − 2·2-s + 2·4-s − 2·5-s − 3·9-s + 4·10-s − 4·16-s + 6·18-s − 2·19-s − 4·20-s − 7·23-s − 6·25-s + 4·29-s + 8·32-s − 6·36-s + 4·38-s + 10·43-s + 6·45-s + 14·46-s − 8·47-s − 10·49-s + 12·50-s − 6·53-s − 8·58-s − 8·64-s − 14·67-s − 16·71-s − 4·73-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s − 9-s + 1.26·10-s − 16-s + 1.41·18-s − 0.458·19-s − 0.894·20-s − 1.45·23-s − 6/5·25-s + 0.742·29-s + 1.41·32-s − 36-s + 0.648·38-s + 1.52·43-s + 0.894·45-s + 2.06·46-s − 1.16·47-s − 1.42·49-s + 1.69·50-s − 0.824·53-s − 1.05·58-s − 64-s − 1.71·67-s − 1.89·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13248\)    =    \(2^{6} \cdot 3^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(0.844703\)
Root analytic conductor: \(0.958685\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 13248,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 8 T + p T^{2} ) \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−10.93258789280890913806509476878, −10.30441496586516194531812282489, −9.878732908970514702821530762431, −9.248540873216968233368807806891, −8.637911197121902682968399668704, −8.243107408854781492365920333522, −7.71141203017232747055351642250, −7.38521952880572566561119236918, −6.26910332953765805561678817235, −6.00219715771373920394908652491, −4.76672620061633582239851058580, −4.11256588402448402034934941925, −3.09711192308674777900480025227, −1.93356732858540601762976651053, 0, 1.93356732858540601762976651053, 3.09711192308674777900480025227, 4.11256588402448402034934941925, 4.76672620061633582239851058580, 6.00219715771373920394908652491, 6.26910332953765805561678817235, 7.38521952880572566561119236918, 7.71141203017232747055351642250, 8.243107408854781492365920333522, 8.637911197121902682968399668704, 9.248540873216968233368807806891, 9.878732908970514702821530762431, 10.30441496586516194531812282489, 10.93258789280890913806509476878

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