Properties

Label 4-1078e2-1.1-c1e2-0-42
Degree $4$
Conductor $1162084$
Sign $1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 4·5-s − 4·6-s + 4·8-s − 9-s − 8·10-s + 2·11-s − 6·12-s − 2·13-s + 8·15-s + 5·16-s − 4·17-s − 2·18-s − 4·19-s − 12·20-s + 4·22-s − 4·23-s − 8·24-s + 4·25-s − 4·26-s + 6·27-s − 6·29-s + 16·30-s − 8·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 1.41·8-s − 1/3·9-s − 2.52·10-s + 0.603·11-s − 1.73·12-s − 0.554·13-s + 2.06·15-s + 5/4·16-s − 0.970·17-s − 0.471·18-s − 0.917·19-s − 2.68·20-s + 0.852·22-s − 0.834·23-s − 1.63·24-s + 4/5·25-s − 0.784·26-s + 1.15·27-s − 1.11·29-s + 2.92·30-s − 1.43·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 18 T + 221 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−10.06477862546213881074250399468, −9.117571181381448836660539112961, −8.647391055893001048566521837963, −8.509975251680844266118807433803, −7.74950471439085965865414281976, −7.46933604672738163711201981400, −6.94669093177188615185586851706, −6.75197710276730297124267104510, −6.08331219643602788103178146321, −5.85494737514462564543108724177, −5.24453842468061287533273340138, −4.92220897291724393041969388405, −4.40137796680275312007265807935, −4.00316208833923154090235521243, −3.42506786528039792244107635968, −3.35509225231927757396983707596, −2.16178137203075410890896305845, −1.78873006098545990279967493870, 0, 0, 1.78873006098545990279967493870, 2.16178137203075410890896305845, 3.35509225231927757396983707596, 3.42506786528039792244107635968, 4.00316208833923154090235521243, 4.40137796680275312007265807935, 4.92220897291724393041969388405, 5.24453842468061287533273340138, 5.85494737514462564543108724177, 6.08331219643602788103178146321, 6.75197710276730297124267104510, 6.94669093177188615185586851706, 7.46933604672738163711201981400, 7.74950471439085965865414281976, 8.509975251680844266118807433803, 8.647391055893001048566521837963, 9.117571181381448836660539112961, 10.06477862546213881074250399468

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