Properties

Degree 2
Conductor $ 3^{2} \cdot 7 $
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 − 7-s + 16-s + 25-s + 28-s − 2·37-s − 2·43-s + 49-s − 64-s + 2·67-s + 2·79-s − 100-s − 2·109-s − 112-s + ⋯
L(s)  = 1  − 4-s − 7-s + 16-s + 25-s + 28-s − 2·37-s − 2·43-s + 49-s − 64-s + 2·67-s + 2·79-s − 100-s − 2·109-s − 112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(63\)    =    \(3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{63} (55, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 63,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4040860762\)
\(L(\frac12)\)  \(\approx\)  \(0.4040860762\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

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

−15.18997093836620602460637582619, −13.99915090221690521549417434165, −13.10575814689347063007868597189, −12.19346500594903014566864998433, −10.47732338749126199986584579156, −9.471054764067809621406584360430, −8.420547362205116289934079288794, −6.74489112464479961545435616225, −5.16255973701201100775027286861, −3.51328813313495541508690022552, 3.51328813313495541508690022552, 5.16255973701201100775027286861, 6.74489112464479961545435616225, 8.420547362205116289934079288794, 9.471054764067809621406584360430, 10.47732338749126199986584579156, 12.19346500594903014566864998433, 13.10575814689347063007868597189, 13.99915090221690521549417434165, 15.18997093836620602460637582619

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