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Properties

Label 4-2368e2-1.1-c0e2-0-1
Degree $4$
Conductor $5607424$
Sign $1$
Analytic cond. $1.39661$
Root an. cond. $1.08709$
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  − 2·7-s + 9-s − 2·17-s − 2·19-s − 2·23-s + 2·29-s + 2·47-s + 49-s − 2·53-s − 2·63-s + 2·71-s − 2·79-s + 2·83-s − 2·89-s + 2·109-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯
L(s)  = 1  − 2·7-s + 9-s − 2·17-s − 2·19-s − 2·23-s + 2·29-s + 2·47-s + 49-s − 2·53-s − 2·63-s + 2·71-s − 2·79-s + 2·83-s − 2·89-s + 2·109-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5607424\)    =    \(2^{12} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(1.39661\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5607424,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5728682947\)
\(L(\frac12)\) \(\approx\) \(0.5728682947\)
\(L(1)\) 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 \( 1 \)
37$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_2^2$ \( 1 - T^{2} + T^{4} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.507444227413182639724242237583, −8.807278911850031532327764001580, −8.793859658400093389506068012217, −8.179978204166365741812017994939, −7.996721827830266799508879916376, −7.15972804931450572252942872587, −7.02315924906918231624602191194, −6.53859801680196118896579963242, −6.23181389123048806553640298027, −6.22626700897869833321045673292, −5.63132072761486984052024572822, −4.67938280251218641218901462148, −4.51788677214061706336735399163, −4.20866552620871849574870979456, −3.72609844272081899159515164610, −3.22779171878161646874349107106, −2.59478174855576320291414965767, −2.20957810832280323988694214502, −1.73348169387667460872441360762, −0.48634662600052422695110422645, 0.48634662600052422695110422645, 1.73348169387667460872441360762, 2.20957810832280323988694214502, 2.59478174855576320291414965767, 3.22779171878161646874349107106, 3.72609844272081899159515164610, 4.20866552620871849574870979456, 4.51788677214061706336735399163, 4.67938280251218641218901462148, 5.63132072761486984052024572822, 6.22626700897869833321045673292, 6.23181389123048806553640298027, 6.53859801680196118896579963242, 7.02315924906918231624602191194, 7.15972804931450572252942872587, 7.996721827830266799508879916376, 8.179978204166365741812017994939, 8.793859658400093389506068012217, 8.807278911850031532327764001580, 9.507444227413182639724242237583

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