Properties

Label 4-1368e2-1.1-c1e2-0-18
Degree $4$
Conductor $1871424$
Sign $-1$
Analytic cond. $119.323$
Root an. cond. $3.30507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 2·11-s − 4·16-s + 2·17-s − 2·19-s − 4·22-s − 25-s − 8·32-s + 4·34-s − 4·38-s − 2·43-s − 4·44-s + 11·49-s − 2·50-s + 16·59-s − 8·64-s + 16·67-s + 4·68-s − 22·73-s − 4·76-s − 24·83-s − 4·86-s + 12·89-s − 20·97-s + 22·98-s − 2·100-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.603·11-s − 16-s + 0.485·17-s − 0.458·19-s − 0.852·22-s − 1/5·25-s − 1.41·32-s + 0.685·34-s − 0.648·38-s − 0.304·43-s − 0.603·44-s + 11/7·49-s − 0.282·50-s + 2.08·59-s − 64-s + 1.95·67-s + 0.485·68-s − 2.57·73-s − 0.458·76-s − 2.63·83-s − 0.431·86-s + 1.27·89-s − 2.03·97-s + 2.22·98-s − 1/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1871424\)    =    \(2^{6} \cdot 3^{4} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(119.323\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1871424,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.46607717556725394550853600860, −6.85073886350288365363080648917, −6.81976651111915002122674082793, −6.12241751609850681792173625996, −5.68404754505961415392399642199, −5.34668146252658900228619106104, −5.12252201851867645807416891194, −4.32311343904703138277137352813, −4.10417783076595364297890095109, −3.67840796781373175574076394185, −2.97029581237660750527806680564, −2.61687998381550726785939862479, −2.09376317548126210224444667076, −1.18431344333758512049731147264, 0, 1.18431344333758512049731147264, 2.09376317548126210224444667076, 2.61687998381550726785939862479, 2.97029581237660750527806680564, 3.67840796781373175574076394185, 4.10417783076595364297890095109, 4.32311343904703138277137352813, 5.12252201851867645807416891194, 5.34668146252658900228619106104, 5.68404754505961415392399642199, 6.12241751609850681792173625996, 6.81976651111915002122674082793, 6.85073886350288365363080648917, 7.46607717556725394550853600860

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