Properties

Label 4-152e2-1.1-c0e2-0-0
Degree $4$
Conductor $23104$
Sign $1$
Analytic cond. $0.00575441$
Root an. cond. $0.275423$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 6-s + 8-s + 9-s − 2·11-s − 16-s − 2·17-s − 18-s − 19-s + 2·22-s + 24-s − 25-s + 2·27-s − 2·33-s + 2·34-s + 38-s + 41-s − 2·43-s − 48-s + 2·49-s + 50-s − 2·51-s − 2·54-s − 57-s + 59-s + 64-s + ⋯
L(s)  = 1  − 2-s + 3-s − 6-s + 8-s + 9-s − 2·11-s − 16-s − 2·17-s − 18-s − 19-s + 2·22-s + 24-s − 25-s + 2·27-s − 2·33-s + 2·34-s + 38-s + 41-s − 2·43-s − 48-s + 2·49-s + 50-s − 2·51-s − 2·54-s − 57-s + 59-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23104\)    =    \(2^{6} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.00575441\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{152} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23104,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3076237393\)
\(L(\frac12)\) \(\approx\) \(0.3076237393\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 + T + T^{2} \)
good3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
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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

−13.25814109219072467176254385597, −13.15459683331786685532889254614, −12.81533261395288879606324937369, −12.02361452398343865595862925576, −10.98704515428475493520638789862, −10.98317146030726192002502408597, −10.12876269430643243027513528919, −10.08373028466783430772051515063, −9.327406285338906020916266976238, −8.657723995035487315844952114077, −8.410114434551979520735328937624, −8.101073976834170344302537445441, −7.18947349529932496515057948897, −7.05485255925857981371645894753, −6.06133272168678729103772022551, −5.05189013214110699027880545501, −4.54406087258176959610073463060, −3.85397393163801438416186496316, −2.56999500303935729279473701049, −2.12233152739616025475607167415, 2.12233152739616025475607167415, 2.56999500303935729279473701049, 3.85397393163801438416186496316, 4.54406087258176959610073463060, 5.05189013214110699027880545501, 6.06133272168678729103772022551, 7.05485255925857981371645894753, 7.18947349529932496515057948897, 8.101073976834170344302537445441, 8.410114434551979520735328937624, 8.657723995035487315844952114077, 9.327406285338906020916266976238, 10.08373028466783430772051515063, 10.12876269430643243027513528919, 10.98317146030726192002502408597, 10.98704515428475493520638789862, 12.02361452398343865595862925576, 12.81533261395288879606324937369, 13.15459683331786685532889254614, 13.25814109219072467176254385597

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