Properties

Label 4-1078e2-1.1-c1e2-0-28
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 2·5-s + 2·6-s − 8-s + 3·9-s + 2·10-s − 11-s + 8·13-s + 4·15-s − 16-s + 3·18-s + 4·19-s − 22-s − 4·23-s − 2·24-s + 5·25-s + 8·26-s + 10·27-s + 4·29-s + 4·30-s − 10·31-s − 2·33-s + 6·37-s + 4·38-s + 16·39-s − 2·40-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.894·5-s + 0.816·6-s − 0.353·8-s + 9-s + 0.632·10-s − 0.301·11-s + 2.21·13-s + 1.03·15-s − 1/4·16-s + 0.707·18-s + 0.917·19-s − 0.213·22-s − 0.834·23-s − 0.408·24-s + 25-s + 1.56·26-s + 1.92·27-s + 0.742·29-s + 0.730·30-s − 1.79·31-s − 0.348·33-s + 0.986·37-s + 0.648·38-s + 2.56·39-s − 0.316·40-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: \(0\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.665019613\)
\(L(\frac12)\) \(\approx\) \(6.665019613\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
11$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01350204916070635451877289179, −9.729518703746625427036580236451, −9.033285076876109913253813217025, −8.769125756712090139796204094030, −8.678046288002148475484347135518, −8.202460298631693284486675252214, −7.39213862833059705124936528491, −7.37919605047910310857874139410, −6.62899288222405712162424905873, −6.09765282408618968103522858111, −5.96189373856331395141331590868, −5.32632710212094652290359697579, −4.93056120075052739828509316368, −4.26311917124467220159868590220, −3.66539088643647559814240604776, −3.60901393889998111002004265296, −2.68805756499205694979615474203, −2.55196958815236995648887714010, −1.53096879039288438458703556727, −1.11876509928895296265418184908, 1.11876509928895296265418184908, 1.53096879039288438458703556727, 2.55196958815236995648887714010, 2.68805756499205694979615474203, 3.60901393889998111002004265296, 3.66539088643647559814240604776, 4.26311917124467220159868590220, 4.93056120075052739828509316368, 5.32632710212094652290359697579, 5.96189373856331395141331590868, 6.09765282408618968103522858111, 6.62899288222405712162424905873, 7.37919605047910310857874139410, 7.39213862833059705124936528491, 8.202460298631693284486675252214, 8.678046288002148475484347135518, 8.769125756712090139796204094030, 9.033285076876109913253813217025, 9.729518703746625427036580236451, 10.01350204916070635451877289179

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