Properties

Label 4-546e2-1.1-c1e2-0-20
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s + 3·11-s − 2·13-s − 14-s − 15-s − 16-s + 2·17-s − 21-s + 3·22-s − 6·23-s − 24-s + 5·25-s − 2·26-s − 27-s + 18·29-s − 30-s − 5·31-s + 3·33-s + 2·34-s + 35-s + 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.904·11-s − 0.554·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s + 25-s − 0.392·26-s − 0.192·27-s + 3.34·29-s − 0.182·30-s − 0.898·31-s + 0.522·33-s + 0.342·34-s + 0.169·35-s + 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.590099897\)
\(L(\frac12)\) \(\approx\) \(2.590099897\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
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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

−10.99263051115799243784621245902, −10.65168223376858963665293482734, −9.969715769515208000355723223381, −9.690002701981892593660266498012, −9.440659719420773425379446678364, −8.503318192651798780981836354118, −8.486052658381911535361474971829, −8.080000712262077456968422914782, −7.36311552992085832989946812460, −6.86882025835806760698301686441, −6.40653738964839307384587513408, −6.14698395014221225722082117442, −5.17728991315345883317191745758, −4.99480724927504952447009825478, −4.09037409148173204379043917726, −4.04653632235164986751997838867, −3.09418854216884818716346871031, −2.90760476207824051830328898095, −1.98424556898645992557921746343, −0.838685091202834542566605555383, 0.838685091202834542566605555383, 1.98424556898645992557921746343, 2.90760476207824051830328898095, 3.09418854216884818716346871031, 4.04653632235164986751997838867, 4.09037409148173204379043917726, 4.99480724927504952447009825478, 5.17728991315345883317191745758, 6.14698395014221225722082117442, 6.40653738964839307384587513408, 6.86882025835806760698301686441, 7.36311552992085832989946812460, 8.080000712262077456968422914782, 8.486052658381911535361474971829, 8.503318192651798780981836354118, 9.440659719420773425379446678364, 9.690002701981892593660266498012, 9.969715769515208000355723223381, 10.65168223376858963665293482734, 10.99263051115799243784621245902

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