Properties

Label 4-720e2-1.1-c1e2-0-32
Degree $4$
Conductor $518400$
Sign $1$
Analytic cond. $33.0536$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s − 4·7-s + 6·9-s + 3·11-s + 4·13-s − 3·15-s + 6·17-s − 10·19-s − 12·21-s + 6·23-s + 9·27-s − 6·29-s + 2·31-s + 9·33-s + 4·35-s − 8·37-s + 12·39-s + 3·41-s + 11·43-s − 6·45-s + 7·49-s + 18·51-s + 12·53-s − 3·55-s − 30·57-s − 3·59-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 1.51·7-s + 2·9-s + 0.904·11-s + 1.10·13-s − 0.774·15-s + 1.45·17-s − 2.29·19-s − 2.61·21-s + 1.25·23-s + 1.73·27-s − 1.11·29-s + 0.359·31-s + 1.56·33-s + 0.676·35-s − 1.31·37-s + 1.92·39-s + 0.468·41-s + 1.67·43-s − 0.894·45-s + 49-s + 2.52·51-s + 1.64·53-s − 0.404·55-s − 3.97·57-s − 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(33.0536\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.208526469\)
\(L(\frac12)\) \(\approx\) \(3.208526469\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49897421605245869236952277247, −10.21289780206880567216753569045, −9.417541308764779902415699585209, −9.353629583556061453203500068952, −8.948955554398129765124944093313, −8.435858603904231763429045115396, −8.344613829901136435493235014713, −7.55870232114401753705904273887, −7.17977942168347120137183831319, −6.88314253294859671971811849541, −6.12212245074065086345496674465, −6.08023312401024454537753387710, −5.17998351658346428593068851196, −4.28873226265988229012981394407, −3.87818085717095173237037694826, −3.63661034085897527631193317907, −3.16098389543368325827477851598, −2.54007572311196241797055661688, −1.83632829658934941938625660347, −0.886081282075011703012248408510, 0.886081282075011703012248408510, 1.83632829658934941938625660347, 2.54007572311196241797055661688, 3.16098389543368325827477851598, 3.63661034085897527631193317907, 3.87818085717095173237037694826, 4.28873226265988229012981394407, 5.17998351658346428593068851196, 6.08023312401024454537753387710, 6.12212245074065086345496674465, 6.88314253294859671971811849541, 7.17977942168347120137183831319, 7.55870232114401753705904273887, 8.344613829901136435493235014713, 8.435858603904231763429045115396, 8.948955554398129765124944093313, 9.353629583556061453203500068952, 9.417541308764779902415699585209, 10.21289780206880567216753569045, 10.49897421605245869236952277247

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