Properties

Label 4-1386e2-1.1-c1e2-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·5-s + 4·7-s − 8-s − 2·10-s − 11-s + 4·13-s + 4·14-s − 16-s − 3·17-s − 7·19-s − 22-s − 7·23-s + 5·25-s + 4·26-s + 10·29-s − 2·31-s − 3·34-s − 8·35-s − 3·37-s − 7·38-s + 2·40-s − 12·41-s + 22·43-s − 7·46-s − 7·47-s + 9·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·5-s + 1.51·7-s − 0.353·8-s − 0.632·10-s − 0.301·11-s + 1.10·13-s + 1.06·14-s − 1/4·16-s − 0.727·17-s − 1.60·19-s − 0.213·22-s − 1.45·23-s + 25-s + 0.784·26-s + 1.85·29-s − 0.359·31-s − 0.514·34-s − 1.35·35-s − 0.493·37-s − 1.13·38-s + 0.316·40-s − 1.87·41-s + 3.35·43-s − 1.03·46-s − 1.02·47-s + 9/7·49-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

\(L(1)\) \(\approx\) \(2.470167400\)
\(L(\frac12)\) \(\approx\) \(2.470167400\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
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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

−9.895705721444759404942649470511, −9.103365845978011882865713404101, −8.864229087123546453339807153033, −8.410169031621487197841065526369, −8.162153498355008181083053194122, −8.021769692244951355312203246700, −7.33121403197587218182134267617, −6.91663142810385347500617003011, −6.30391235869713457883867170346, −6.18558141707468322921965351351, −5.54443811769757547149259025990, −4.97362124359990972677763714903, −4.55422991061390050910049718472, −4.43643586564463960772723892275, −3.76282872200635909759487602606, −3.57117051866448721690388977488, −2.62413673213178740729403676099, −2.18846439817452727018719664461, −1.52641916201463872689615329654, −0.58397585232842752378855491801, 0.58397585232842752378855491801, 1.52641916201463872689615329654, 2.18846439817452727018719664461, 2.62413673213178740729403676099, 3.57117051866448721690388977488, 3.76282872200635909759487602606, 4.43643586564463960772723892275, 4.55422991061390050910049718472, 4.97362124359990972677763714903, 5.54443811769757547149259025990, 6.18558141707468322921965351351, 6.30391235869713457883867170346, 6.91663142810385347500617003011, 7.33121403197587218182134267617, 8.021769692244951355312203246700, 8.162153498355008181083053194122, 8.410169031621487197841065526369, 8.864229087123546453339807153033, 9.103365845978011882865713404101, 9.895705721444759404942649470511

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