Properties

Degree 2
Conductor $ 2 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s − 13-s − 2·14-s − 15-s + 16-s + 8·17-s − 2·18-s + 20-s + 2·21-s − 3·22-s + 4·23-s − 24-s − 4·25-s − 26-s + 5·27-s − 2·28-s − 29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.471·18-s + 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.962·27-s − 0.377·28-s − 0.185·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(58\)    =    \(2 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{58} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 58,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.033207509$
$L(\frac12)$  $\approx$  $1.033207509$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
29 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−19.38317527311314, −18.30617017141159, −16.98213068342467, −16.38511723856289, −15.09322040009075, −13.99719858770620, −12.92256133309153, −12.00978753036483, −10.74800585012116, −9.633495492381884, −7.714361855637271, −6.116589998588637, −5.251367906235375, −3.070718811356957, 3.070718811356957, 5.251367906235375, 6.116589998588637, 7.714361855637271, 9.633495492381884, 10.74800585012116, 12.00978753036483, 12.92256133309153, 13.99719858770620, 15.09322040009075, 16.38511723856289, 16.98213068342467, 18.30617017141159, 19.38317527311314

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