Properties

Label 4-48e4-1.1-c1e2-0-2
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $338.469$
Root an. cond. $4.28923$
Motivic weight $1$
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  + 4·7-s − 12·17-s − 8·23-s + 10·25-s − 20·31-s − 4·41-s − 24·47-s − 2·49-s + 8·71-s + 20·73-s + 12·79-s + 4·89-s − 12·97-s − 20·103-s + 28·113-s − 48·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s − 2.91·17-s − 1.66·23-s + 2·25-s − 3.59·31-s − 0.624·41-s − 3.50·47-s − 2/7·49-s + 0.949·71-s + 2.34·73-s + 1.35·79-s + 0.423·89-s − 1.21·97-s − 1.97·103-s + 2.63·113-s − 4.40·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 2.52·161-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(338.469\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.067515487\)
\(L(\frac12)\) \(\approx\) \(1.067515487\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.347326338096017142438418932643, −8.552507796422888236654309077914, −8.534186416259772447699450274897, −8.154492849090770594003693589834, −7.76083197203837316097170918019, −7.23822972083437491706903627209, −6.84879269632144457025565923293, −6.46911112005204211609772978174, −6.32367353764861735640204142475, −5.47744472932571929058752740041, −5.08068589332762480492005333298, −4.88059365848098470921519903672, −4.55539523296991156612474715791, −3.81354758989909082739341563029, −3.70820004112830324720460019691, −2.91502579540998992668842195212, −2.19758761057679051211430770345, −1.75287322298532242533148896277, −1.70345657787929674360977736431, −0.32950532247544385902251709809, 0.32950532247544385902251709809, 1.70345657787929674360977736431, 1.75287322298532242533148896277, 2.19758761057679051211430770345, 2.91502579540998992668842195212, 3.70820004112830324720460019691, 3.81354758989909082739341563029, 4.55539523296991156612474715791, 4.88059365848098470921519903672, 5.08068589332762480492005333298, 5.47744472932571929058752740041, 6.32367353764861735640204142475, 6.46911112005204211609772978174, 6.84879269632144457025565923293, 7.23822972083437491706903627209, 7.76083197203837316097170918019, 8.154492849090770594003693589834, 8.534186416259772447699450274897, 8.552507796422888236654309077914, 9.347326338096017142438418932643

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