Properties

Degree 2
Conductor $ 11 \cdot 233 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 9-s + 11-s + 16-s − 17-s − 23-s + 25-s − 31-s + 36-s − 37-s − 41-s − 43-s + 44-s + 49-s + 2·61-s + 64-s − 68-s − 71-s − 73-s − 79-s + 81-s + 2·83-s − 89-s − 92-s + 99-s + 100-s − 113-s + ⋯
L(s)  = 1  + 4-s + 9-s + 11-s + 16-s − 17-s − 23-s + 25-s − 31-s + 36-s − 37-s − 41-s − 43-s + 44-s + 49-s + 2·61-s + 64-s − 68-s − 71-s − 73-s − 79-s + 81-s + 2·83-s − 89-s − 92-s + 99-s + 100-s − 113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2563\)    =    \(11 \cdot 233\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2563} (2562, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2563,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.704728343$
$L(\frac12)$  $\approx$  $1.704728343$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{11,\;233\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{11,\;233\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad11 \( 1 - T \)
233 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( 1 + T + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.020009334974855580080227092243, −8.347176406388379027602919459600, −7.23446647040270288310065959559, −6.89934430819586723881572927115, −6.22587729353245931673220085239, −5.20827325966413228781924762719, −4.17336978762595771348959176719, −3.44817324685758271131182164983, −2.18914478852139758103090914772, −1.43634349865073593314751131417, 1.43634349865073593314751131417, 2.18914478852139758103090914772, 3.44817324685758271131182164983, 4.17336978762595771348959176719, 5.20827325966413228781924762719, 6.22587729353245931673220085239, 6.89934430819586723881572927115, 7.23446647040270288310065959559, 8.347176406388379027602919459600, 9.020009334974855580080227092243

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