Properties

Label 4-7200e2-1.1-c1e2-0-52
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 $0$

Origins

Origins of factors

Downloads

Learn more

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: \(0\)
Selberg data: \((4,\ 51840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.759566818\)
\(L(\frac12)\) \(\approx\) \(5.759566818\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−8.096559439968840208930334390025, −7.83635595286014022258095023805, −7.38196287531328426459859164677, −7.33821538064769685917881881518, −6.55304506873024720074428570896, −6.45660213987737243046529437775, −5.94103262162492931038179789181, −5.67834446320787553307869061323, −5.07503917300434549897758671992, −4.87903609808944360686114794223, −4.58767631266704864393775510878, −4.40062423072192247189511174106, −3.76033628270095452825522072513, −3.44818114533106543883266761275, −2.79107918921604666135985678376, −2.46129744287166072400837008542, −1.84434307512047331684423106027, −1.75800995098082718900107585275, −0.869168304842140774857568928108, −0.75755906389362835296778853913, 0.75755906389362835296778853913, 0.869168304842140774857568928108, 1.75800995098082718900107585275, 1.84434307512047331684423106027, 2.46129744287166072400837008542, 2.79107918921604666135985678376, 3.44818114533106543883266761275, 3.76033628270095452825522072513, 4.40062423072192247189511174106, 4.58767631266704864393775510878, 4.87903609808944360686114794223, 5.07503917300434549897758671992, 5.67834446320787553307869061323, 5.94103262162492931038179789181, 6.45660213987737243046529437775, 6.55304506873024720074428570896, 7.33821538064769685917881881518, 7.38196287531328426459859164677, 7.83635595286014022258095023805, 8.096559439968840208930334390025

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