Properties

Label 4-7200e2-1.1-c1e2-0-56
Degree $4$
Conductor $51840000$
Sign $1$
Analytic cond. $3305.36$
Root an. cond. $7.58236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s + 2·23-s + 12·29-s − 18·43-s + 14·47-s + 18·49-s − 6·67-s − 22·83-s − 12·89-s + 36·101-s − 18·103-s − 26·107-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 12·161-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.26·7-s + 0.417·23-s + 2.22·29-s − 2.74·43-s + 2.04·47-s + 18/7·49-s − 0.733·67-s − 2.41·83-s − 1.27·89-s + 3.58·101-s − 1.77·103-s − 2.51·107-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.945·161-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(51840000\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3305.36\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 51840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−7.62081365392883341695482804143, −7.33786911872680985945907246697, −6.82036308005866578813864802442, −6.80134433986183344720185227076, −6.35431746478519814786174703435, −6.21087200142691704897560517434, −5.54283558990614559770365988999, −5.54033214653268332821198739754, −4.75199757081376341618382502286, −4.66346077894810481427661498659, −3.98873067187532353551688351596, −3.76343590414311702510030460514, −3.13094899512245181238412380782, −3.12556403686703036773009843628, −2.54623329728980688922779759892, −2.31924727511527629839171557918, −1.27936797021892842346315534408, −1.11164591770651581612138279584, 0, 0, 1.11164591770651581612138279584, 1.27936797021892842346315534408, 2.31924727511527629839171557918, 2.54623329728980688922779759892, 3.12556403686703036773009843628, 3.13094899512245181238412380782, 3.76343590414311702510030460514, 3.98873067187532353551688351596, 4.66346077894810481427661498659, 4.75199757081376341618382502286, 5.54033214653268332821198739754, 5.54283558990614559770365988999, 6.21087200142691704897560517434, 6.35431746478519814786174703435, 6.80134433986183344720185227076, 6.82036308005866578813864802442, 7.33786911872680985945907246697, 7.62081365392883341695482804143

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