Properties

Label 4-62720-1.1-c1e2-0-0
Degree $4$
Conductor $62720$
Sign $1$
Analytic cond. $3.99908$
Root an. cond. $1.41413$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 3·7-s + 9-s − 11-s − 3·13-s − 2·25-s + 11·31-s + 6·35-s + 4·43-s + 2·45-s + 12·47-s + 2·49-s − 2·55-s + 8·61-s + 3·63-s − 6·65-s − 10·67-s − 3·77-s − 8·81-s − 9·91-s − 99-s − 12·101-s + 4·103-s − 2·107-s + 10·113-s − 3·117-s − 9·121-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s − 2/5·25-s + 1.97·31-s + 1.01·35-s + 0.609·43-s + 0.298·45-s + 1.75·47-s + 2/7·49-s − 0.269·55-s + 1.02·61-s + 0.377·63-s − 0.744·65-s − 1.22·67-s − 0.341·77-s − 8/9·81-s − 0.943·91-s − 0.100·99-s − 1.19·101-s + 0.394·103-s − 0.193·107-s + 0.940·113-s − 0.277·117-s − 0.818·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62720 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62720 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(62720\)    =    \(2^{8} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.99908\)
Root analytic conductor: \(1.41413\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 62720,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.860930439\)
\(L(\frac12)\) \(\approx\) \(1.860930439\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 84 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 67 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
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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

−9.966231345866097962223326346795, −9.539518008497246779238566668643, −8.886146178488241912387094791902, −8.434580261867451508271976627939, −7.81659797349164421818623801817, −7.46594384674791843729086651713, −6.84504662223972423976334453644, −6.14530080650181788179357213479, −5.66345287248134878346872655815, −5.04705364593051019346087334191, −4.56462790641652387474444789252, −3.95155491908117360313674434808, −2.73407040905016519565170316268, −2.24868565115079375992838713356, −1.27112817255976606343847966295, 1.27112817255976606343847966295, 2.24868565115079375992838713356, 2.73407040905016519565170316268, 3.95155491908117360313674434808, 4.56462790641652387474444789252, 5.04705364593051019346087334191, 5.66345287248134878346872655815, 6.14530080650181788179357213479, 6.84504662223972423976334453644, 7.46594384674791843729086651713, 7.81659797349164421818623801817, 8.434580261867451508271976627939, 8.886146178488241912387094791902, 9.539518008497246779238566668643, 9.966231345866097962223326346795

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