Properties

Label 4-1216800-1.1-c1e2-0-21
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 + 4·5-s + 8-s + 9-s + 4·10-s − 2·13-s + 16-s + 2·17-s + 18-s + 4·20-s + 11·25-s − 2·26-s − 2·29-s + 32-s + 2·34-s + 36-s + 4·40-s + 2·41-s + 4·45-s − 2·49-s + 11·50-s − 2·52-s + 16·53-s − 2·58-s + 18·61-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.894·20-s + 11/5·25-s − 0.392·26-s − 0.371·29-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.632·40-s + 0.312·41-s + 0.596·45-s − 2/7·49-s + 1.55·50-s − 0.277·52-s + 2.19·53-s − 0.262·58-s + 2.30·61-s + 1/8·64-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

\(L(1)\) \(\approx\) \(5.449397471\)
\(L(\frac12)\) \(\approx\) \(5.449397471\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \)
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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

−8.001722285573530234099699587513, −7.34011520358803540240866276358, −7.03463286180866588865982538203, −6.69429331963496380064397628388, −6.10954320097515553702613587615, −5.78744901221085719000645416904, −5.31131637869523665115613024574, −5.11592734284799164705999992717, −4.46590563837048033633908516960, −3.91084097552295410765159416752, −3.36754075604026682444152849530, −2.54053427454284915376231308857, −2.38932205054278202703595782696, −1.67005249900893672816398613375, −0.974895579872811435981273565160, 0.974895579872811435981273565160, 1.67005249900893672816398613375, 2.38932205054278202703595782696, 2.54053427454284915376231308857, 3.36754075604026682444152849530, 3.91084097552295410765159416752, 4.46590563837048033633908516960, 5.11592734284799164705999992717, 5.31131637869523665115613024574, 5.78744901221085719000645416904, 6.10954320097515553702613587615, 6.69429331963496380064397628388, 7.03463286180866588865982538203, 7.34011520358803540240866276358, 8.001722285573530234099699587513

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