Properties

Label 4-1344e2-1.1-c0e2-0-4
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $0.449896$
Root an. cond. $0.818989$
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  + 3-s − 7-s + 2·13-s − 19-s − 21-s − 25-s − 27-s + 31-s − 37-s + 2·39-s + 2·43-s − 57-s + 2·61-s − 67-s + 73-s − 75-s + 79-s − 81-s − 2·91-s + 93-s + 4·97-s + 103-s − 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯
L(s)  = 1  + 3-s − 7-s + 2·13-s − 19-s − 21-s − 25-s − 27-s + 31-s − 37-s + 2·39-s + 2·43-s − 57-s + 2·61-s − 67-s + 73-s − 75-s + 79-s − 81-s − 2·91-s + 93-s + 4·97-s + 103-s − 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.449896\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359520331\)
\(L(\frac12)\) \(\approx\) \(1.359520331\)
\(L(1)\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_1$ \( ( 1 - T )^{4} \)
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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

−9.997954344333303524717599043405, −9.383467158233869330309434519483, −9.170360850273264025598008120593, −8.808915666656241617420891102977, −8.506554621931060771096713992486, −7.960070583648503875698738693212, −7.86080495463675631444327463042, −7.19652795889189431779836154334, −6.53153640417336782669873061070, −6.46804651908342530034393502210, −5.86860027522183108490775221435, −5.68546210133623193923663978823, −4.91505695290780064439623235451, −4.20038556684155557208127900518, −3.88051996524180013920209145277, −3.45768450465414803479217889066, −3.11227141670397092014620725843, −2.32619378445995974936277107524, −2.00816874465665383006964055389, −0.971873684487282232698546411294, 0.971873684487282232698546411294, 2.00816874465665383006964055389, 2.32619378445995974936277107524, 3.11227141670397092014620725843, 3.45768450465414803479217889066, 3.88051996524180013920209145277, 4.20038556684155557208127900518, 4.91505695290780064439623235451, 5.68546210133623193923663978823, 5.86860027522183108490775221435, 6.46804651908342530034393502210, 6.53153640417336782669873061070, 7.19652795889189431779836154334, 7.86080495463675631444327463042, 7.960070583648503875698738693212, 8.506554621931060771096713992486, 8.808915666656241617420891102977, 9.170360850273264025598008120593, 9.383467158233869330309434519483, 9.997954344333303524717599043405

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