Properties

Label 4-1680e2-1.1-c1e2-0-17
Degree $4$
Conductor $2822400$
Sign $1$
Analytic cond. $179.958$
Root an. cond. $3.66263$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 9-s + 12·11-s − 12·19-s − 25-s + 4·29-s + 20·31-s + 4·41-s − 2·45-s − 49-s + 24·55-s − 16·59-s − 4·61-s − 20·71-s + 8·79-s + 81-s − 12·89-s − 24·95-s − 12·99-s − 12·101-s − 4·109-s + 86·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s + 3.61·11-s − 2.75·19-s − 1/5·25-s + 0.742·29-s + 3.59·31-s + 0.624·41-s − 0.298·45-s − 1/7·49-s + 3.23·55-s − 2.08·59-s − 0.512·61-s − 2.37·71-s + 0.900·79-s + 1/9·81-s − 1.27·89-s − 2.46·95-s − 1.20·99-s − 1.19·101-s − 0.383·109-s + 7.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(179.958\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.546623820\)
\(L(\frac12)\) \(\approx\) \(3.546623820\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.630484987504833201068314081556, −9.037316624036424670548798856841, −8.901764698100648976381902966375, −8.391286020574861295884542478247, −8.275276814594937286758130429113, −7.55394851584204516514435058663, −6.89254801942769031751785061203, −6.41465917000920879765685209947, −6.40801056846433653809037194446, −6.22190877898798408188190991752, −5.77297122144178884552119185260, −4.83581668636817029977509134717, −4.37051167475054357166874939529, −4.25095807035757933035455119348, −3.86335002827774524147219868801, −2.93737312533058031418294316443, −2.68373142689175862657521282022, −1.65001263413998875062560435512, −1.63339636286715631086532618820, −0.75683000651717450731875337317, 0.75683000651717450731875337317, 1.63339636286715631086532618820, 1.65001263413998875062560435512, 2.68373142689175862657521282022, 2.93737312533058031418294316443, 3.86335002827774524147219868801, 4.25095807035757933035455119348, 4.37051167475054357166874939529, 4.83581668636817029977509134717, 5.77297122144178884552119185260, 6.22190877898798408188190991752, 6.40801056846433653809037194446, 6.41465917000920879765685209947, 6.89254801942769031751785061203, 7.55394851584204516514435058663, 8.275276814594937286758130429113, 8.391286020574861295884542478247, 8.901764698100648976381902966375, 9.037316624036424670548798856841, 9.630484987504833201068314081556

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