Properties

Label 4-2736e2-1.1-c0e2-0-0
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $1.86443$
Root an. cond. $1.16852$
Motivic weight $0$
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  − 2·7-s − 3·13-s + 2·19-s + 25-s − 43-s + 49-s − 61-s − 3·67-s + 73-s + 3·79-s + 6·91-s − 2·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·7-s − 3·13-s + 2·19-s + 25-s − 43-s + 49-s − 61-s − 3·67-s + 73-s + 3·79-s + 6·91-s − 2·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.86443\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5529633901\)
\(L(\frac12)\) \(\approx\) \(0.5529633901\)
\(L(1)\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−9.441745388027253689956561372730, −8.995895180678542776277115577399, −8.537485418341376265690210624144, −7.81681491718672206284192146434, −7.53189804481116670919883617897, −7.43020527753381586090306463394, −6.91681211065318024499585023462, −6.52321417514903310232637302742, −6.34826758260039717661456176861, −5.71516146990152639360945575936, −5.17710922816259131839983574731, −4.99497728007132371328790400023, −4.67648375550538696526344388330, −3.96229600867309390172410672302, −3.38553863517960802531946724868, −3.01845737590230745737867451544, −2.83421409205258426244237504865, −2.29035475215703533085928477681, −1.49569383600972657544422535565, −0.46855008322279034826699165570, 0.46855008322279034826699165570, 1.49569383600972657544422535565, 2.29035475215703533085928477681, 2.83421409205258426244237504865, 3.01845737590230745737867451544, 3.38553863517960802531946724868, 3.96229600867309390172410672302, 4.67648375550538696526344388330, 4.99497728007132371328790400023, 5.17710922816259131839983574731, 5.71516146990152639360945575936, 6.34826758260039717661456176861, 6.52321417514903310232637302742, 6.91681211065318024499585023462, 7.43020527753381586090306463394, 7.53189804481116670919883617897, 7.81681491718672206284192146434, 8.537485418341376265690210624144, 8.995895180678542776277115577399, 9.441745388027253689956561372730

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