Properties

Label 4-1386e2-1.1-c1e2-0-1
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-s − 5·7-s + 8-s − 10-s + 11-s + 4·13-s + 5·14-s − 16-s − 5·17-s + 6·19-s − 22-s − 7·23-s + 5·25-s − 4·26-s − 16·29-s − 10·31-s + 5·34-s − 5·35-s + 8·37-s − 6·38-s + 40-s + 14·41-s + 8·43-s + 7·46-s + 47-s + 18·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s − 1.88·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 1.10·13-s + 1.33·14-s − 1/4·16-s − 1.21·17-s + 1.37·19-s − 0.213·22-s − 1.45·23-s + 25-s − 0.784·26-s − 2.97·29-s − 1.79·31-s + 0.857·34-s − 0.845·35-s + 1.31·37-s − 0.973·38-s + 0.158·40-s + 2.18·41-s + 1.21·43-s + 1.03·46-s + 0.145·47-s + 18/7·49-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

\(L(1)\) \(\approx\) \(0.6641299167\)
\(L(\frac12)\) \(\approx\) \(0.6641299167\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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

−9.851439868380208394489644934443, −9.329089323766612776114813287527, −9.021264943123153085011227062122, −8.955934164237614169862937553666, −8.302214666357111984571158427533, −7.49938594597107300060627663411, −7.35418970922135567860569920951, −7.24135324232164606746908589139, −6.37979775787264892846013391132, −5.99174435709958435634642022734, −5.83285281244924969927521519655, −5.58151589743541318394326982153, −4.44020669765414236443630727933, −4.26194123883535688924601137073, −3.58219645814744840782560104815, −3.33501319882675610036449611930, −2.61276875405228630328051999727, −2.01563993210522905304771047129, −1.32251688516892974383587393045, −0.39173942278724470151872436545, 0.39173942278724470151872436545, 1.32251688516892974383587393045, 2.01563993210522905304771047129, 2.61276875405228630328051999727, 3.33501319882675610036449611930, 3.58219645814744840782560104815, 4.26194123883535688924601137073, 4.44020669765414236443630727933, 5.58151589743541318394326982153, 5.83285281244924969927521519655, 5.99174435709958435634642022734, 6.37979775787264892846013391132, 7.24135324232164606746908589139, 7.35418970922135567860569920951, 7.49938594597107300060627663411, 8.302214666357111984571158427533, 8.955934164237614169862937553666, 9.021264943123153085011227062122, 9.329089323766612776114813287527, 9.851439868380208394489644934443

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