Properties

Label 4-477603-1.1-c1e2-0-15
Degree $4$
Conductor $477603$
Sign $1$
Analytic cond. $30.4523$
Root an. cond. $2.34912$
Motivic weight $1$
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 + 6·5-s − 5·7-s + 9-s + 6·15-s − 4·16-s + 2·17-s − 5·21-s + 17·25-s + 27-s − 30·35-s − 2·43-s + 6·45-s + 18·47-s − 4·48-s + 18·49-s + 2·51-s + 16·59-s − 5·63-s + 16·67-s + 17·75-s + 32·79-s − 24·80-s + 81-s − 24·83-s + 12·85-s + 12·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.68·5-s − 1.88·7-s + 1/3·9-s + 1.54·15-s − 16-s + 0.485·17-s − 1.09·21-s + 17/5·25-s + 0.192·27-s − 5.07·35-s − 0.304·43-s + 0.894·45-s + 2.62·47-s − 0.577·48-s + 18/7·49-s + 0.280·51-s + 2.08·59-s − 0.629·63-s + 1.95·67-s + 1.96·75-s + 3.60·79-s − 2.68·80-s + 1/9·81-s − 2.63·83-s + 1.30·85-s + 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(477603\)    =    \(3^{3} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(30.4523\)
Root analytic conductor: \(2.34912\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 477603,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.212156944\)
\(L(\frac12)\) \(\approx\) \(3.212156944\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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, −8.240904623337692395017705850814, −7.50551633645791376109359922005, −6.85073886350288365363080648917, −6.77741729329520342893941173390, −6.12241751609850681792173625996, −5.95880974340205553779958707744, −5.34668146252658900228619106104, −5.02464724211845145360498207900, −3.92167943470614491379352796776, −3.67840796781373175574076394185, −2.61687998381550726785939862479, −2.54192725156556063621144024653, −1.98299256211429317844363710869, −0.930874084897434565967858870729, 0.930874084897434565967858870729, 1.98299256211429317844363710869, 2.54192725156556063621144024653, 2.61687998381550726785939862479, 3.67840796781373175574076394185, 3.92167943470614491379352796776, 5.02464724211845145360498207900, 5.34668146252658900228619106104, 5.95880974340205553779958707744, 6.12241751609850681792173625996, 6.77741729329520342893941173390, 6.85073886350288365363080648917, 7.50551633645791376109359922005, 8.240904623337692395017705850814, 8.959163091125505196869204673189

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