Properties

Label 4-4608e2-1.1-c1e2-0-2
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  − 2·5-s − 8·11-s − 6·13-s + 12·17-s − 8·19-s + 2·25-s − 6·29-s − 8·31-s − 2·37-s − 8·43-s − 16·47-s − 2·49-s + 14·53-s + 16·55-s + 6·61-s + 12·65-s + 16·67-s + 24·79-s + 8·83-s − 24·85-s + 16·95-s + 16·97-s + 14·101-s + 16·107-s − 10·109-s + 32·113-s + 32·121-s + ⋯
L(s)  = 1  − 0.894·5-s − 2.41·11-s − 1.66·13-s + 2.91·17-s − 1.83·19-s + 2/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 1.21·43-s − 2.33·47-s − 2/7·49-s + 1.92·53-s + 2.15·55-s + 0.768·61-s + 1.48·65-s + 1.95·67-s + 2.70·79-s + 0.878·83-s − 2.60·85-s + 1.64·95-s + 1.62·97-s + 1.39·101-s + 1.54·107-s − 0.957·109-s + 3.01·113-s + 2.90·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2906914859\)
\(L(\frac12)\) \(\approx\) \(0.2906914859\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−8.191369991531178345946684644724, −8.154808575879200940249911699795, −7.78084216713438401172447844650, −7.47202288082686122165464303569, −7.26109435703064281738303446419, −6.88745631777288707806945643988, −6.27919015280741253125612102299, −5.84959340781190739597332303173, −5.38621865391246481105874663365, −5.19892344305830342994375772856, −4.80274980257673117290925887284, −4.69908980103553041460351177319, −3.64102982532909511144476535342, −3.52311399073376054035589778596, −3.40430811391255203648354803734, −2.56870128143738965139144512185, −2.14798075190753806272401614586, −1.98774875785737705277861269324, −0.872140762584174519217697791786, −0.17979818392754076906837527253, 0.17979818392754076906837527253, 0.872140762584174519217697791786, 1.98774875785737705277861269324, 2.14798075190753806272401614586, 2.56870128143738965139144512185, 3.40430811391255203648354803734, 3.52311399073376054035589778596, 3.64102982532909511144476535342, 4.69908980103553041460351177319, 4.80274980257673117290925887284, 5.19892344305830342994375772856, 5.38621865391246481105874663365, 5.84959340781190739597332303173, 6.27919015280741253125612102299, 6.88745631777288707806945643988, 7.26109435703064281738303446419, 7.47202288082686122165464303569, 7.78084216713438401172447844650, 8.154808575879200940249911699795, 8.191369991531178345946684644724

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