Properties

Label 4-1134e2-1.1-c1e2-0-4
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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 − 10·11-s + 4·14-s − 16-s − 4·17-s − 8·19-s − 10·22-s + 8·23-s − 7·25-s − 5·29-s − 3·31-s − 4·34-s − 8·35-s + 4·37-s − 8·38-s + 2·40-s − 2·43-s + 8·46-s − 6·47-s + 9·49-s − 7·50-s − 9·53-s + 20·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·5-s + 1.51·7-s − 0.353·8-s − 0.632·10-s − 3.01·11-s + 1.06·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 2.13·22-s + 1.66·23-s − 7/5·25-s − 0.928·29-s − 0.538·31-s − 0.685·34-s − 1.35·35-s + 0.657·37-s − 1.29·38-s + 0.316·40-s − 0.304·43-s + 1.17·46-s − 0.875·47-s + 9/7·49-s − 0.989·50-s − 1.23·53-s + 2.69·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1134} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5436923444\)
\(L(\frac12)\) \(\approx\) \(0.5436923444\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−10.33561829894124781275833330163, −9.539724846620569600839619587997, −9.193709751118144522342267709656, −8.455097941681831741228575242327, −8.350707281662787806307649899356, −7.85306818704118945017810025630, −7.83576576311369556179647744827, −7.05555626716470911464647289123, −6.94158105914127859257004100464, −5.84497064628903122383256838000, −5.78255094467497370465565471221, −5.22168614633782176726790053995, −4.73545741747289084490952996941, −4.40424962104469368739024281006, −4.25235097662830384124164097681, −3.20587130788720384140379930745, −2.97184717759455502003556207532, −2.04739439191801017158504118707, −1.93831245664780331247692642475, −0.26708513409587534597085687755, 0.26708513409587534597085687755, 1.93831245664780331247692642475, 2.04739439191801017158504118707, 2.97184717759455502003556207532, 3.20587130788720384140379930745, 4.25235097662830384124164097681, 4.40424962104469368739024281006, 4.73545741747289084490952996941, 5.22168614633782176726790053995, 5.78255094467497370465565471221, 5.84497064628903122383256838000, 6.94158105914127859257004100464, 7.05555626716470911464647289123, 7.83576576311369556179647744827, 7.85306818704118945017810025630, 8.350707281662787806307649899356, 8.455097941681831741228575242327, 9.193709751118144522342267709656, 9.539724846620569600839619587997, 10.33561829894124781275833330163

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