Properties

Label 4-3724e2-1.1-c0e2-0-2
Degree $4$
Conductor $13868176$
Sign $1$
Analytic cond. $3.45408$
Root an. cond. $1.36327$
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  + 5-s − 9-s + 11-s + 17-s − 19-s − 2·23-s + 25-s − 2·43-s − 45-s + 47-s + 55-s + 61-s + 73-s + 4·83-s + 85-s − 95-s − 99-s − 2·101-s − 2·115-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 5-s − 9-s + 11-s + 17-s − 19-s − 2·23-s + 25-s − 2·43-s − 45-s + 47-s + 55-s + 61-s + 73-s + 4·83-s + 85-s − 95-s − 99-s − 2·101-s − 2·115-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13868176\)    =    \(2^{4} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3.45408\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3724} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13868176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.595807069\)
\(L(\frac12)\) \(\approx\) \(1.595807069\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
19$C_2$ \( 1 + T + T^{2} \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.717109917227330457579181591885, −8.699629564000920235757333711119, −8.124982455574306957142741798460, −7.909236072443285415643004601586, −7.50783802004575073445301186780, −6.81166801410978967905108111328, −6.48957887246313796201305956739, −6.43768228897130868959087639998, −5.85600528650101313618957763664, −5.61881413269342878491488954252, −5.23116276040291331517536910185, −4.81419628247920123479889106032, −4.23782174999867611170028495022, −3.81565440144836153143760433219, −3.47791715399893294117677873613, −2.97264276023836715606033146379, −2.34453309674846738681748061847, −2.01209835935887637317845673145, −1.56922280159680543184160473022, −0.72593903728398656606338243250, 0.72593903728398656606338243250, 1.56922280159680543184160473022, 2.01209835935887637317845673145, 2.34453309674846738681748061847, 2.97264276023836715606033146379, 3.47791715399893294117677873613, 3.81565440144836153143760433219, 4.23782174999867611170028495022, 4.81419628247920123479889106032, 5.23116276040291331517536910185, 5.61881413269342878491488954252, 5.85600528650101313618957763664, 6.43768228897130868959087639998, 6.48957887246313796201305956739, 6.81166801410978967905108111328, 7.50783802004575073445301186780, 7.909236072443285415643004601586, 8.124982455574306957142741798460, 8.699629564000920235757333711119, 8.717109917227330457579181591885

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