Properties

Label 4-966e2-1.1-c1e2-0-0
Degree $4$
Conductor $933156$
Sign $1$
Analytic cond. $59.4988$
Root an. cond. $2.77732$
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 + 3-s + 3·5-s − 6-s − 5·7-s + 8-s − 3·10-s − 11-s − 12·13-s + 5·14-s + 3·15-s − 16-s + 2·17-s + 2·19-s − 5·21-s + 22-s + 23-s + 24-s + 5·25-s + 12·26-s − 27-s − 14·29-s − 3·30-s − 7·31-s − 33-s − 2·34-s − 15·35-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1.34·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s − 0.948·10-s − 0.301·11-s − 3.32·13-s + 1.33·14-s + 0.774·15-s − 1/4·16-s + 0.485·17-s + 0.458·19-s − 1.09·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s + 25-s + 2.35·26-s − 0.192·27-s − 2.59·29-s − 0.547·30-s − 1.25·31-s − 0.174·33-s − 0.342·34-s − 2.53·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(933156\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(59.4988\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{966} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 933156,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3218244612\)
\(L(\frac12)\) \(\approx\) \(0.3218244612\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
23$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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

−10.05589631496281501368192555717, −9.676213546611871581085748678867, −9.487620842230493111988887035186, −9.270344512506887448638965920088, −8.870372306342280414195622581316, −8.150304588394075340463508576525, −7.48257196277459954240382371188, −7.37049190507704357889919229485, −7.14320863162329243498013726139, −6.30777994158210965966384176577, −6.15555771762743253557862360701, −5.34602186115786228687311210255, −5.09353705955192793856188714465, −4.76075201385361770602888148257, −3.50795430069871278115690057824, −3.42561085167064956658946670582, −2.76383080759142530179728279853, −2.04828039071573666403744488370, −1.92237527958281162874417283040, −0.26067682897197828543551700295, 0.26067682897197828543551700295, 1.92237527958281162874417283040, 2.04828039071573666403744488370, 2.76383080759142530179728279853, 3.42561085167064956658946670582, 3.50795430069871278115690057824, 4.76075201385361770602888148257, 5.09353705955192793856188714465, 5.34602186115786228687311210255, 6.15555771762743253557862360701, 6.30777994158210965966384176577, 7.14320863162329243498013726139, 7.37049190507704357889919229485, 7.48257196277459954240382371188, 8.150304588394075340463508576525, 8.870372306342280414195622581316, 9.270344512506887448638965920088, 9.487620842230493111988887035186, 9.676213546611871581085748678867, 10.05589631496281501368192555717

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