Properties

Label 4-3840e2-1.1-c1e2-0-70
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 4·17-s − 8·23-s − 25-s + 12·41-s − 24·47-s − 14·49-s − 4·73-s − 16·79-s + 81-s − 4·89-s − 28·97-s − 16·103-s + 12·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  − 1/3·9-s − 0.970·17-s − 1.66·23-s − 1/5·25-s + 1.87·41-s − 3.50·47-s − 2·49-s − 0.468·73-s − 1.80·79-s + 1/9·81-s − 0.423·89-s − 2.84·97-s − 1.57·103-s + 1.12·113-s + 6/11·121-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 + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−8.142655882146513320848952342960, −8.075263460359479750511363664537, −7.72973860925260609962130087477, −7.14738697888463370612283972642, −6.79167009903293050975870440860, −6.43615873796510427765755664625, −6.10319062723245459059555705664, −5.80949005153059272425789435923, −5.28245199162719436119870350016, −4.90191953837499255577179687411, −4.43293941909340970940938807943, −4.15276760460354368331489310057, −3.69499445721257335797393812289, −3.15224091191776120944315956507, −2.74001181536765894131453375515, −2.28082523926521492288051502464, −1.66091355616582563257249936183, −1.32706305741639967784361735240, 0, 0, 1.32706305741639967784361735240, 1.66091355616582563257249936183, 2.28082523926521492288051502464, 2.74001181536765894131453375515, 3.15224091191776120944315956507, 3.69499445721257335797393812289, 4.15276760460354368331489310057, 4.43293941909340970940938807943, 4.90191953837499255577179687411, 5.28245199162719436119870350016, 5.80949005153059272425789435923, 6.10319062723245459059555705664, 6.43615873796510427765755664625, 6.79167009903293050975870440860, 7.14738697888463370612283972642, 7.72973860925260609962130087477, 8.075263460359479750511363664537, 8.142655882146513320848952342960

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