Properties

Label 4-3840e2-1.1-c1e2-0-35
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s + 3·9-s − 8·15-s + 11·25-s + 4·27-s + 8·31-s − 16·37-s + 20·41-s − 8·43-s − 12·45-s − 2·49-s + 24·53-s + 8·67-s + 22·75-s + 24·79-s + 5·81-s + 8·83-s − 20·89-s + 16·93-s + 24·107-s − 32·111-s + 6·121-s + 40·123-s − 24·125-s + 127-s − 16·129-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s − 2.06·15-s + 11/5·25-s + 0.769·27-s + 1.43·31-s − 2.63·37-s + 3.12·41-s − 1.21·43-s − 1.78·45-s − 2/7·49-s + 3.29·53-s + 0.977·67-s + 2.54·75-s + 2.70·79-s + 5/9·81-s + 0.878·83-s − 2.11·89-s + 1.65·93-s + 2.32·107-s − 3.03·111-s + 6/11·121-s + 3.60·123-s − 2.14·125-s + 0.0887·127-s − 1.40·129-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

\(L(1)\) \(\approx\) \(3.148072913\)
\(L(\frac12)\) \(\approx\) \(3.148072913\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.438889038947173182413323343330, −8.426220001072337156911335852935, −7.965815070039479399631035195020, −7.68141460742279706607448359768, −7.25741095001970878006871203103, −7.02004650485077925950007305841, −6.67788901736851606326597663012, −6.26345118242283070785353508383, −5.51051806295697786845614915780, −5.30929832302428030216489175673, −4.59340973585300201209166650320, −4.47135175044073417687056955176, −3.94434004355725885132051536689, −3.63925703361201978622807868801, −3.31941608057261962450531382365, −2.83003538722097492246331937631, −2.34943906245832997555044402730, −1.89069860782359118672668043211, −0.955018373414104480512377092868, −0.59737789044940422032016358078, 0.59737789044940422032016358078, 0.955018373414104480512377092868, 1.89069860782359118672668043211, 2.34943906245832997555044402730, 2.83003538722097492246331937631, 3.31941608057261962450531382365, 3.63925703361201978622807868801, 3.94434004355725885132051536689, 4.47135175044073417687056955176, 4.59340973585300201209166650320, 5.30929832302428030216489175673, 5.51051806295697786845614915780, 6.26345118242283070785353508383, 6.67788901736851606326597663012, 7.02004650485077925950007305841, 7.25741095001970878006871203103, 7.68141460742279706607448359768, 7.965815070039479399631035195020, 8.426220001072337156911335852935, 8.438889038947173182413323343330

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