Properties

Label 4-1216800-1.1-c1e2-0-6
Degree $4$
Conductor $1216800$
Sign $1$
Analytic cond. $77.5842$
Root an. cond. $2.96785$
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-s + 4-s − 2·5-s + 8-s − 9-s − 2·10-s − 2·13-s + 16-s − 18-s − 2·20-s + 3·25-s − 2·26-s + 8·29-s + 32-s − 36-s − 4·37-s − 2·40-s + 8·41-s + 2·45-s + 2·49-s + 3·50-s − 2·52-s − 8·53-s + 8·58-s − 12·61-s + 64-s + 4·65-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 1/3·9-s − 0.632·10-s − 0.554·13-s + 1/4·16-s − 0.235·18-s − 0.447·20-s + 3/5·25-s − 0.392·26-s + 1.48·29-s + 0.176·32-s − 1/6·36-s − 0.657·37-s − 0.316·40-s + 1.24·41-s + 0.298·45-s + 2/7·49-s + 0.424·50-s − 0.277·52-s − 1.09·53-s + 1.05·58-s − 1.53·61-s + 1/8·64-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1216800\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(77.5842\)
Root analytic conductor: \(2.96785\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1216800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.233578078\)
\(L(\frac12)\) \(\approx\) \(2.233578078\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_2$ \( 1 + T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.974329083861169461711204688509, −7.60596310537835363074242785647, −7.09520691033161893312951010813, −6.71766181336806969605447659541, −6.24686353651114259833355695102, −5.83547941932160596416963661520, −5.23079364960260719714520326457, −4.77585675339976825744446972892, −4.47476552731699880082383495806, −3.92876495927149337247602505891, −3.36025638556076581595314614050, −2.90534186805765197938165483521, −2.41815729124572029272394317986, −1.57105541181048644388291882397, −0.59545077410238119269402403979, 0.59545077410238119269402403979, 1.57105541181048644388291882397, 2.41815729124572029272394317986, 2.90534186805765197938165483521, 3.36025638556076581595314614050, 3.92876495927149337247602505891, 4.47476552731699880082383495806, 4.77585675339976825744446972892, 5.23079364960260719714520326457, 5.83547941932160596416963661520, 6.24686353651114259833355695102, 6.71766181336806969605447659541, 7.09520691033161893312951010813, 7.60596310537835363074242785647, 7.974329083861169461711204688509

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