Properties

Label 4-399e2-1.1-c1e2-0-7
Degree $4$
Conductor $159201$
Sign $-1$
Analytic cond. $10.1507$
Root an. cond. $1.78494$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 5·7-s − 8·8-s + 9-s + 2·11-s + 20·14-s − 4·16-s − 4·18-s − 8·22-s − 8·23-s − 25-s − 40·28-s − 4·29-s + 32·32-s + 8·36-s − 2·43-s + 16·44-s + 32·46-s + 18·49-s + 4·50-s + 20·53-s + 40·56-s + 16·58-s − 5·63-s − 64·64-s + 16·67-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 1.88·7-s − 2.82·8-s + 1/3·9-s + 0.603·11-s + 5.34·14-s − 16-s − 0.942·18-s − 1.70·22-s − 1.66·23-s − 1/5·25-s − 7.55·28-s − 0.742·29-s + 5.65·32-s + 4/3·36-s − 0.304·43-s + 2.41·44-s + 4.71·46-s + 18/7·49-s + 0.565·50-s + 2.74·53-s + 5.34·56-s + 2.10·58-s − 0.629·63-s − 8·64-s + 1.95·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(159201\)    =    \(3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(10.1507\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 159201,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.959163091125505196869204673189, −8.836427381447004052087018681826, −8.240904623337692395017705850814, −7.50551633645791376109359922005, −7.47848778675296589128891353641, −6.77741729329520342893941173390, −6.41173993136661837405396202601, −5.95880974340205553779958707744, −5.02464724211845145360498207900, −3.92167943470614491379352796776, −3.81591641511052084615077615666, −2.54192725156556063621144024653, −1.98299256211429317844363710869, −0.930874084897434565967858870729, 0, 0.930874084897434565967858870729, 1.98299256211429317844363710869, 2.54192725156556063621144024653, 3.81591641511052084615077615666, 3.92167943470614491379352796776, 5.02464724211845145360498207900, 5.95880974340205553779958707744, 6.41173993136661837405396202601, 6.77741729329520342893941173390, 7.47848778675296589128891353641, 7.50551633645791376109359922005, 8.240904623337692395017705850814, 8.836427381447004052087018681826, 8.959163091125505196869204673189

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