Properties

Label 4-1200e2-1.1-c2e2-0-21
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $1069.13$
Root an. cond. $5.71818$
Motivic weight $2$
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  + 3-s + 16·7-s − 8·9-s − 4·13-s − 22·19-s + 16·21-s − 17·27-s + 92·31-s + 32·37-s − 4·39-s + 124·43-s + 94·49-s − 22·57-s − 32·61-s − 128·63-s + 226·67-s − 202·73-s − 136·79-s + 55·81-s − 64·91-s + 92·93-s + 44·97-s + 52·103-s + 352·109-s + 32·111-s + 32·117-s − 73·121-s + ⋯
L(s)  = 1  + 1/3·3-s + 16/7·7-s − 8/9·9-s − 0.307·13-s − 1.15·19-s + 0.761·21-s − 0.629·27-s + 2.96·31-s + 0.864·37-s − 0.102·39-s + 2.88·43-s + 1.91·49-s − 0.385·57-s − 0.524·61-s − 2.03·63-s + 3.37·67-s − 2.76·73-s − 1.72·79-s + 0.679·81-s − 0.703·91-s + 0.989·93-s + 0.453·97-s + 0.504·103-s + 3.22·109-s + 0.288·111-s + 0.273·117-s − 0.603·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1069.13\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.178125235\)
\(L(\frac12)\) \(\approx\) \(4.178125235\)
\(L(2)\) 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$C_2$ \( 1 - T + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 29 T + p^{2} T^{2} )( 1 + 29 T + p^{2} T^{2} ) \)
19$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 202 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 422 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 527 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3158 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4358 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 1922 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 113 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 1258 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 101 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 13463 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 13007 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{2} \)
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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

−9.850593898804046294539661728391, −9.220392785961744204865643940803, −8.729308228141334049619387225024, −8.445714136487085271556949721215, −8.291703415065667236211511152864, −7.69802484254808182178168583935, −7.59903130782252000962467945800, −6.98276980673596953145490801971, −6.16244070696292573157067135268, −6.13352289307853883002477016007, −5.51654733592879906091759237707, −4.91749018845847083820293719435, −4.57271142789805412096121521407, −4.34345649447911130584429053018, −3.72404214635462295647506257984, −2.74543782384356575426931918215, −2.56127366772225805873811755341, −2.00170341477063596083510680171, −1.25478643712709726228180034542, −0.63629568886352041505168016092, 0.63629568886352041505168016092, 1.25478643712709726228180034542, 2.00170341477063596083510680171, 2.56127366772225805873811755341, 2.74543782384356575426931918215, 3.72404214635462295647506257984, 4.34345649447911130584429053018, 4.57271142789805412096121521407, 4.91749018845847083820293719435, 5.51654733592879906091759237707, 6.13352289307853883002477016007, 6.16244070696292573157067135268, 6.98276980673596953145490801971, 7.59903130782252000962467945800, 7.69802484254808182178168583935, 8.291703415065667236211511152864, 8.445714136487085271556949721215, 8.729308228141334049619387225024, 9.220392785961744204865643940803, 9.850593898804046294539661728391

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