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  =  $(2,\ 2563,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.625656111$
$L(\frac12)$  $\approx$  $1.625656111$
$L(1)$   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

−9.138238770019637857515807416091, −8.038455575902191871532525256564, −7.50655293335190340913384381869, −6.95972156521946188203131360537, −5.95665714033469672537204693292, −5.34484361578186438437618562053, −4.25131015883942217649791022222, −3.27003321731519927151931412343, −2.35914235718737809731244368492, −1.34360636371185014023045938662, 1.34360636371185014023045938662, 2.35914235718737809731244368492, 3.27003321731519927151931412343, 4.25131015883942217649791022222, 5.34484361578186438437618562053, 5.95665714033469672537204693292, 6.95972156521946188203131360537, 7.50655293335190340913384381869, 8.038455575902191871532525256564, 9.138238770019637857515807416091

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