Properties

Label 4-4560e2-1.1-c1e2-0-2
Degree $4$
Conductor $20793600$
Sign $1$
Analytic cond. $1325.81$
Root an. cond. $6.03421$
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  − 2·3-s − 2·5-s + 4·7-s + 3·9-s + 4·15-s + 2·19-s − 8·21-s + 4·23-s + 3·25-s − 4·27-s − 4·29-s + 12·31-s − 8·35-s − 4·41-s + 12·43-s − 6·45-s − 4·47-s − 8·53-s − 4·57-s − 16·61-s + 12·63-s − 8·69-s + 16·71-s + 4·73-s − 6·75-s + 16·79-s + 5·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1.51·7-s + 9-s + 1.03·15-s + 0.458·19-s − 1.74·21-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s + 2.15·31-s − 1.35·35-s − 0.624·41-s + 1.82·43-s − 0.894·45-s − 0.583·47-s − 1.09·53-s − 0.529·57-s − 2.04·61-s + 1.51·63-s − 0.963·69-s + 1.89·71-s + 0.468·73-s − 0.692·75-s + 1.80·79-s + 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20793600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1325.81\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20793600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.957746351\)
\(L(\frac12)\) \(\approx\) \(1.957746351\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T - 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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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

−8.232734835395951866786952815297, −8.051770561901685873738901575430, −7.73372965530860845693000267229, −7.66953936516252925853522840146, −6.91076218640348467049664668994, −6.80101487378957457616175655600, −6.19425871892741556483277878447, −6.15199153696383977971176696767, −5.27235523143529734396562364263, −5.23025417682103031654133567100, −4.73638373518931480150586112572, −4.69661079306401521636875259251, −3.95329200373950509739396774569, −3.89446926439897743578843011575, −3.01785667320687823454346050223, −2.83864412546053812854048244951, −1.94355958571798899954313066482, −1.58748057196747311901682976967, −0.937794634178985329137266693806, −0.53727142922983314127949709546, 0.53727142922983314127949709546, 0.937794634178985329137266693806, 1.58748057196747311901682976967, 1.94355958571798899954313066482, 2.83864412546053812854048244951, 3.01785667320687823454346050223, 3.89446926439897743578843011575, 3.95329200373950509739396774569, 4.69661079306401521636875259251, 4.73638373518931480150586112572, 5.23025417682103031654133567100, 5.27235523143529734396562364263, 6.15199153696383977971176696767, 6.19425871892741556483277878447, 6.80101487378957457616175655600, 6.91076218640348467049664668994, 7.66953936516252925853522840146, 7.73372965530860845693000267229, 8.051770561901685873738901575430, 8.232734835395951866786952815297

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