Properties

Label 4-162e2-1.1-c1e2-0-9
Degree $4$
Conductor $26244$
Sign $-1$
Analytic cond. $1.67334$
Root an. cond. $1.13735$
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  + 4-s − 8·7-s − 2·13-s + 16-s − 8·19-s − 25-s − 8·28-s − 8·31-s − 2·37-s + 16·43-s + 34·49-s − 2·52-s − 2·61-s + 64-s − 8·67-s + 22·73-s − 8·76-s − 32·79-s + 16·91-s + 4·97-s − 100-s − 8·103-s + 22·109-s − 8·112-s − 22·121-s − 8·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 3.02·7-s − 0.554·13-s + 1/4·16-s − 1.83·19-s − 1/5·25-s − 1.51·28-s − 1.43·31-s − 0.328·37-s + 2.43·43-s + 34/7·49-s − 0.277·52-s − 0.256·61-s + 1/8·64-s − 0.977·67-s + 2.57·73-s − 0.917·76-s − 3.60·79-s + 1.67·91-s + 0.406·97-s − 0.0999·100-s − 0.788·103-s + 2.10·109-s − 0.755·112-s − 2·121-s − 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26244\)    =    \(2^{2} \cdot 3^{8}\)
Sign: $-1$
Analytic conductor: \(1.67334\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{26244} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.31837968175541511748482298781, −9.880068252452829064500103261921, −9.247376955218132439005872939948, −9.101601246278505892226541271616, −8.278948375699901432058512087738, −7.24345462277249174674167705434, −7.16277862972928710155421440227, −6.29261335708839240986736404399, −6.18735538198950166736525445007, −5.49090131952205180218546644094, −4.25638665319375785757127060430, −3.69589098584017718744945020548, −2.94651413139897301734292877776, −2.30016225881827778465495235815, 0, 2.30016225881827778465495235815, 2.94651413139897301734292877776, 3.69589098584017718744945020548, 4.25638665319375785757127060430, 5.49090131952205180218546644094, 6.18735538198950166736525445007, 6.29261335708839240986736404399, 7.16277862972928710155421440227, 7.24345462277249174674167705434, 8.278948375699901432058512087738, 9.101601246278505892226541271616, 9.247376955218132439005872939948, 9.880068252452829064500103261921, 10.31837968175541511748482298781

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