Properties

Label 4-1386e2-1.1-c1e2-0-41
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·5-s − 5·7-s − 8-s + 2·10-s − 11-s − 14·13-s − 5·14-s − 16-s + 2·17-s − 22-s − 8·23-s + 5·25-s − 14·26-s + 10·29-s − 4·31-s + 2·34-s − 10·35-s − 4·37-s − 2·40-s − 8·41-s − 16·43-s − 8·46-s + 2·47-s + 18·49-s + 5·50-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.894·5-s − 1.88·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 3.88·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.213·22-s − 1.66·23-s + 25-s − 2.74·26-s + 1.85·29-s − 0.718·31-s + 0.342·34-s − 1.69·35-s − 0.657·37-s − 0.316·40-s − 1.24·41-s − 2.43·43-s − 1.17·46-s + 0.291·47-s + 18/7·49-s + 0.707·50-s − 0.824·53-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: \(2\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

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)$
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$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−9.463651737599295965675675668980, −9.347146940890400031167254519453, −8.402711167269090113813159075923, −8.393748599256950342727763959778, −7.38053985947741429936379499473, −7.35142333821652578905366075368, −6.87195576413003465722387076695, −6.35447981703526983183437500642, −6.22151945108275359559471990606, −5.43014031734927876090125655462, −5.15038466116120683249202141835, −4.91386442592581204627499716929, −4.34586621938657870757983730193, −3.68612678543057485420691162952, −3.06702934538345590599517576196, −2.73874047689079527364115532002, −2.41477780911271363056326392541, −1.67638202708494261981625072922, 0, 0, 1.67638202708494261981625072922, 2.41477780911271363056326392541, 2.73874047689079527364115532002, 3.06702934538345590599517576196, 3.68612678543057485420691162952, 4.34586621938657870757983730193, 4.91386442592581204627499716929, 5.15038466116120683249202141835, 5.43014031734927876090125655462, 6.22151945108275359559471990606, 6.35447981703526983183437500642, 6.87195576413003465722387076695, 7.35142333821652578905366075368, 7.38053985947741429936379499473, 8.393748599256950342727763959778, 8.402711167269090113813159075923, 9.347146940890400031167254519453, 9.463651737599295965675675668980

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