Properties

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

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 10-s − 11-s − 12-s − 13-s + 15-s + 16-s + 2·19-s − 20-s − 22-s − 24-s − 26-s + 27-s + 29-s + 30-s − 31-s + 32-s + 33-s + 2·38-s + 39-s − 40-s − 43-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 10-s − 11-s − 12-s − 13-s + 15-s + 16-s + 2·19-s − 20-s − 22-s − 24-s − 26-s + 27-s + 29-s + 30-s − 31-s + 32-s + 33-s + 2·38-s + 39-s − 40-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(116\)    =    \(2^{2} \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{116} (115, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 116,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6792266740\)
\(L(\frac12)\)  \(\approx\)  \(0.6792266740\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

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

−13.74254106518691585642797536546, −12.49738155406116875572269725213, −11.84093943455898711558667245967, −11.16700118547112250006640331579, −10.04581052862782411520590533496, −7.927650309728446968100608044272, −7.05540575330027098655761742099, −5.56304821134120925869298444888, −4.78764434142051405584396595400, −3.11568281765995191104117732931, 3.11568281765995191104117732931, 4.78764434142051405584396595400, 5.56304821134120925869298444888, 7.05540575330027098655761742099, 7.927650309728446968100608044272, 10.04581052862782411520590533496, 11.16700118547112250006640331579, 11.84093943455898711558667245967, 12.49738155406116875572269725213, 13.74254106518691585642797536546

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