Properties

Label 4-1386e2-1.1-c1e2-0-29
Degree $4$
Conductor $1920996$
Sign $1$
Analytic cond. $122.484$
Root an. cond. $3.32675$
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-s + 5·7-s − 8-s − 11-s + 10·13-s + 5·14-s − 16-s + 6·17-s − 2·19-s − 22-s + 6·23-s + 5·25-s + 10·26-s − 6·29-s − 8·31-s + 6·34-s − 2·37-s − 2·38-s + 12·41-s − 8·43-s + 6·46-s + 6·47-s + 18·49-s + 5·50-s − 12·53-s − 5·56-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.88·7-s − 0.353·8-s − 0.301·11-s + 2.77·13-s + 1.33·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.213·22-s + 1.25·23-s + 25-s + 1.96·26-s − 1.11·29-s − 1.43·31-s + 1.02·34-s − 0.328·37-s − 0.324·38-s + 1.87·41-s − 1.21·43-s + 0.884·46-s + 0.875·47-s + 18/7·49-s + 0.707·50-s − 1.64·53-s − 0.668·56-s − 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(122.484\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.402103255\)
\(L(\frac12)\) \(\approx\) \(5.402103255\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−9.542029117643362321591924420731, −9.379555336746172130787091316693, −8.889022219218410515695836530303, −8.451100725055314291334004970982, −8.147951823831937719370681748093, −7.961537165579687276118566942626, −7.36251084211016189474548588857, −6.89413141030597965906877147854, −6.41716003807162379629165287649, −5.89666689105492550792566417797, −5.44033758188113459763450083063, −5.21335681472076505683043377906, −4.91270848811148986621272277109, −4.05808537982217627440062105224, −3.82410473994110530795284801153, −3.50870654016409637107085711208, −2.76390606311101058541979669854, −2.04283183714581064365589456438, −1.27505483400045629054360020248, −1.06698112310847881631634707623, 1.06698112310847881631634707623, 1.27505483400045629054360020248, 2.04283183714581064365589456438, 2.76390606311101058541979669854, 3.50870654016409637107085711208, 3.82410473994110530795284801153, 4.05808537982217627440062105224, 4.91270848811148986621272277109, 5.21335681472076505683043377906, 5.44033758188113459763450083063, 5.89666689105492550792566417797, 6.41716003807162379629165287649, 6.89413141030597965906877147854, 7.36251084211016189474548588857, 7.961537165579687276118566942626, 8.147951823831937719370681748093, 8.451100725055314291334004970982, 8.889022219218410515695836530303, 9.379555336746172130787091316693, 9.542029117643362321591924420731

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