Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 13 $
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  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 3·7-s + 9-s + 2·10-s − 11-s − 2·12-s − 13-s + 6·14-s − 15-s − 4·16-s − 17-s + 2·18-s − 2·19-s + 2·20-s − 3·21-s − 2·22-s − 3·23-s + 25-s − 2·26-s − 27-s + 6·28-s − 2·29-s − 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 1.60·14-s − 0.258·15-s − 16-s − 0.242·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.654·21-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.13·28-s − 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(195\)    =    \(3 \cdot 5 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{195} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 195,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.121203277$
$L(\frac12)$  $\approx$  $2.121203277$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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

−19.92669664369854, −18.39198618548817, −17.91293474388736, −16.92851826196903, −15.93949682395238, −14.83798905852058, −14.44206320285004, −13.32481910848841, −12.71270080469208, −11.61771491043825, −11.06044491519638, −9.771886360481557, −8.335023576107244, −6.970570678349363, −5.793751201825021, −5.054289531359601, −4.051631845302499, −2.211089068348616, 2.211089068348616, 4.051631845302499, 5.054289531359601, 5.793751201825021, 6.970570678349363, 8.335023576107244, 9.771886360481557, 11.06044491519638, 11.61771491043825, 12.71270080469208, 13.32481910848841, 14.44206320285004, 14.83798905852058, 15.93949682395238, 16.92851826196903, 17.91293474388736, 18.39198618548817, 19.92669664369854

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