Properties

Label 4-456e2-1.1-c1e2-0-26
Degree $4$
Conductor $207936$
Sign $1$
Analytic cond. $13.2581$
Root an. cond. $1.90818$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 6·5-s + 9-s + 4·12-s + 12·15-s + 4·16-s + 2·19-s + 12·20-s + 17·25-s + 4·27-s − 12·29-s − 2·36-s − 2·43-s − 6·45-s + 6·47-s − 8·48-s − 13·49-s − 24·53-s − 4·57-s − 24·60-s − 8·64-s − 8·67-s − 12·71-s − 14·73-s − 34·75-s − 4·76-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 2.68·5-s + 1/3·9-s + 1.15·12-s + 3.09·15-s + 16-s + 0.458·19-s + 2.68·20-s + 17/5·25-s + 0.769·27-s − 2.22·29-s − 1/3·36-s − 0.304·43-s − 0.894·45-s + 0.875·47-s − 1.15·48-s − 1.85·49-s − 3.29·53-s − 0.529·57-s − 3.09·60-s − 64-s − 0.977·67-s − 1.42·71-s − 1.63·73-s − 3.92·75-s − 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(207936\)    =    \(2^{6} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(13.2581\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 207936,\ (\ :1/2, 1/2),\ 1)\)

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

−8.462914393396165009068265832341, −7.968751144610877316457317025190, −7.65894969899749366452649043261, −7.43141403312000422638829977686, −6.68601933176938591059893570713, −6.07700564140158959836320975863, −5.51097905283736056328679977781, −4.96903618429312636799150610379, −4.41466206651142691425988858089, −4.19306407725294924458030580762, −3.33866387034377157662854676826, −3.23857552173567549926694025031, −1.39430917521719853589747801849, 0, 0, 1.39430917521719853589747801849, 3.23857552173567549926694025031, 3.33866387034377157662854676826, 4.19306407725294924458030580762, 4.41466206651142691425988858089, 4.96903618429312636799150610379, 5.51097905283736056328679977781, 6.07700564140158959836320975863, 6.68601933176938591059893570713, 7.43141403312000422638829977686, 7.65894969899749366452649043261, 7.968751144610877316457317025190, 8.462914393396165009068265832341

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