Properties

Label 4-3840e2-1.1-c1e2-0-29
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 $0$

Origins

Origins of factors

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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: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.527660092\)
\(L(\frac12)\) \(\approx\) \(2.527660092\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.697402441538386350238848742775, −8.319125142455782124072716279994, −8.002363420704153567279291899771, −7.46042777954692204712799623731, −7.24508239503512494838217655427, −6.85838879498153382910980243468, −6.54979762922825881444686422125, −5.96278195874314239303295123858, −5.73294573375495546384283384345, −5.43742737680556945262430087325, −4.66803738592479078461342966742, −4.65064048627579760159166614043, −4.15657122838561286427890235461, −3.63134600332045456906158320176, −3.16054607192989109082607551240, −2.70368152227799927533369374741, −2.32824668932091512799559347122, −1.79947100943392382824594943396, −1.00478179388458173121284552246, −0.54422233506717417579270614218, 0.54422233506717417579270614218, 1.00478179388458173121284552246, 1.79947100943392382824594943396, 2.32824668932091512799559347122, 2.70368152227799927533369374741, 3.16054607192989109082607551240, 3.63134600332045456906158320176, 4.15657122838561286427890235461, 4.65064048627579760159166614043, 4.66803738592479078461342966742, 5.43742737680556945262430087325, 5.73294573375495546384283384345, 5.96278195874314239303295123858, 6.54979762922825881444686422125, 6.85838879498153382910980243468, 7.24508239503512494838217655427, 7.46042777954692204712799623731, 8.002363420704153567279291899771, 8.319125142455782124072716279994, 8.697402441538386350238848742775

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