Properties

Degree 2
Conductor 83
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 − 7-s − 11-s − 12-s + 16-s − 17-s + 21-s + 2·23-s + 25-s + 27-s − 28-s − 29-s − 31-s + 33-s − 37-s + 2·41-s − 44-s − 48-s + 51-s − 59-s − 61-s + 64-s − 68-s − 2·69-s − 75-s + 77-s + ⋯
L(s)  = 1  − 3-s + 4-s − 7-s − 11-s − 12-s + 16-s − 17-s + 21-s + 2·23-s + 25-s + 27-s − 28-s − 29-s − 31-s + 33-s − 37-s + 2·41-s − 44-s − 48-s + 51-s − 59-s − 61-s + 64-s − 68-s − 2·69-s − 75-s + 77-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{83} (82, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 83,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4493265963$
$L(\frac12)$  $\approx$  $0.4493265963$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 83$, \(F_p\) is a polynomial of degree 2. If $p = 83$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad83 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( 1 + T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
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

−14.86356454031015697333911315034, −13.09589613086253146464545743202, −12.43990908860202224240837164010, −11.01446604629011781551544529494, −10.76555010249983891945786170955, −9.121647327898758088329847107513, −7.30510568839843872365517009292, −6.37890239293184761621864982629, −5.25394517311578314477026475826, −2.91606813106134337451054950253, 2.91606813106134337451054950253, 5.25394517311578314477026475826, 6.37890239293184761621864982629, 7.30510568839843872365517009292, 9.121647327898758088329847107513, 10.76555010249983891945786170955, 11.01446604629011781551544529494, 12.43990908860202224240837164010, 13.09589613086253146464545743202, 14.86356454031015697333911315034

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