Properties

Degree 2
Conductor 59
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 5-s − 7-s − 12-s + 15-s + 16-s + 2·17-s − 19-s − 20-s + 21-s + 27-s − 28-s − 29-s + 35-s − 41-s − 48-s − 2·51-s − 53-s + 57-s + 59-s + 60-s + 64-s + 2·68-s + 2·71-s − 76-s − 79-s + ⋯
L(s)  = 1  − 3-s + 4-s − 5-s − 7-s − 12-s + 15-s + 16-s + 2·17-s − 19-s − 20-s + 21-s + 27-s − 28-s − 29-s + 35-s − 41-s − 48-s − 2·51-s − 53-s + 57-s + 59-s + 60-s + 64-s + 2·68-s + 2·71-s − 76-s − 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{59} (58, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 59,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3770726116\)
\(L(\frac12)\)  \(\approx\)  \(0.3770726116\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 59$,\(F_p(T)\) is a polynomial of degree 2. If $p = 59$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad59 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( 1 + T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )^{2} \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.69721150385094496767899354800, −14.63420773928722559629961047382, −12.66665942627658830350838610678, −11.96978648192178724260299450099, −11.09915165564731068328520576306, −10.00684138086693597518137898757, −7.988751469884320746824006089085, −6.74805173897083286463010363905, −5.64723355410045232763599810849, −3.43180500410537741355091128030, 3.43180500410537741355091128030, 5.64723355410045232763599810849, 6.74805173897083286463010363905, 7.988751469884320746824006089085, 10.00684138086693597518137898757, 11.09915165564731068328520576306, 11.96978648192178724260299450099, 12.66665942627658830350838610678, 14.63420773928722559629961047382, 15.69721150385094496767899354800

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