Properties

Label 4-797-1.1-c1e2-0-0
Degree $4$
Conductor $797$
Sign $1$
Analytic cond. $0.0508174$
Root an. cond. $0.474791$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 7-s + 9-s − 3·11-s + 2·12-s − 13-s + 7·17-s + 3·19-s + 21-s − 2·25-s − 4·27-s + 2·28-s + 7·29-s + 3·33-s − 2·36-s + 37-s + 39-s − 5·41-s − 3·43-s + 6·44-s − 8·47-s − 11·49-s − 7·51-s + 2·52-s + 7·53-s − 3·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 0.277·13-s + 1.69·17-s + 0.688·19-s + 0.218·21-s − 2/5·25-s − 0.769·27-s + 0.377·28-s + 1.29·29-s + 0.522·33-s − 1/3·36-s + 0.164·37-s + 0.160·39-s − 0.780·41-s − 0.457·43-s + 0.904·44-s − 1.16·47-s − 1.57·49-s − 0.980·51-s + 0.277·52-s + 0.961·53-s − 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 797 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 797 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(797\)
Sign: $1$
Analytic conductor: \(0.0508174\)
Root analytic conductor: \(0.474791\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 797,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3559385607\)
\(L(\frac12)\) \(\approx\) \(0.3559385607\)
\(L(\frac{3}{2})\) 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)$
bad797$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 14 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 40 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T - 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 19 T + 208 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T - 27 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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

−19.6679227232, −19.1687844395, −18.5945819642, −18.0669742721, −17.7774940465, −16.9725635965, −16.4191061415, −15.9379138107, −15.2040931962, −14.4650318890, −13.7923572937, −13.3009950214, −12.6609675324, −12.0008038338, −11.3934994195, −10.2212626601, −10.0011394250, −9.24056418157, −8.19123900334, −7.59780337154, −6.45288273711, −5.40128604449, −4.79821105212, −3.37269439426, 3.37269439426, 4.79821105212, 5.40128604449, 6.45288273711, 7.59780337154, 8.19123900334, 9.24056418157, 10.0011394250, 10.2212626601, 11.3934994195, 12.0008038338, 12.6609675324, 13.3009950214, 13.7923572937, 14.4650318890, 15.2040931962, 15.9379138107, 16.4191061415, 16.9725635965, 17.7774940465, 18.0669742721, 18.5945819642, 19.1687844395, 19.6679227232

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